Aquaponic System Blueprints

Aquaponics combine hydroponics (soil-less growing of the plants) and aquaculture (fish farm). Acid and pool chlorine are fine as they are easy to wash out with water. We spent yesterday afternoon finishing the greenhouse. Flowers begin appearing when the chile plant starts branching and the process of flowering is called “dichotomous”, meaning that the plant produces one flower, then two, then four, eight, sixteen and so on.   the pipe could be about 12” or so long to start with. Clean the cup or ring regularly. Setting up the renogy solar system was extremely easy. How many and how big of tanks for you think we need. We explain this more in planning a commercial aquaponics greenhouse. To get a better sense of the best equipment and practices for beginning and intermediate freshwater-aquarium owners, i spoke with several experts for guidance.

aquaponic system blueprints
aquaponic system blueprints

Aquaponics tips: commercial aquaponics usa.   different parts and mechanisms working together to provide mutual support and to sustain a balance. Demand in china and india has outstripped the possible supply by such a margin globally that almost half the fish consumed is produced / farmed in aquaculture. Do you intend to rear the fish for consumption or purely for ornamental purpose. There are many challenges that you are facing if you want to design and build great aquaponic system for the best results and yields. Grow beds are watered with water from the rainbow trout tank via a pipe. The process includes the breakdown of fish waste into plant nutrients, the uptake of these nutrients by the plants being grown in separate grow beds and the cleaning of the water to be returned to the fish tank to be reused in an endless loop.  thought that was a great way to go until it was time to put fish in.

aquaponic system blueprints
aquaponic system blueprints

It combines the use of hydroponics-the. Algae don't grow for without a reason. A guy i met used to do tomatoes and cucumbers with minnows and while i'd like to go with talapia (yum) having it in the front yard on a city street has me erring on the side of caution. While peat moss is naturally-occurring, overharvesting has damaged or destroyed bogs in many areas—coir is a more sustainable alternative. The second piece of information needed is going to be based on what you choose for 2. It also will not taste anything like fresh, locally grown shrimp harvested an hour or so ago. We believe the way to overcome high food prices, the concern of chemically treated and genetically modified foods, and the scare of future food scarcity is to simply grow your own food. Hydroponics fish farm – diy aquaponics gardening combines the best parts of hydroponics and aquaculture. It may grow fish well, but the proteins, fats and carbs in mammal meat are very different from those in fish or vegetables. The other productivity issue arises out of the heightened risk of operating an aquaponics system outside of the optimum water quality parameters.

aquaponic system blueprints
aquaponic system blueprints

Vertical scrog systems don’t always have the plants themselves raised vertically. Turn on the sprinkler on a hot summer day, rainbow on the bamboo grove. There is too much emphasis on ph in aquaponics in my opinion. At this moment though it is about 2 feet above where they work in a symbiotic environmentally all about keeping you more sensitive ones that goes into. For the first time in history,. Currently expanding into aquaculture, they are likely to include aquaponics in the next couple of years.

aquaponic system blueprints
aquaponic system blueprints

Right here you will uncover some web pages that we assume…. Endurance filter: when not to drain the sludge or how to avoid bead loss in your endurance filter. Shui stood back under the stars in the vancouver aquarium in many parts of the spectrum retirement holding tank. Your local hatchery could tell you for sure. And then multiply the number of kilowatt-hours used by the cost of a kilowatt-hour to know the cost to run your light for that number of hours. These days, my time has shifted away from our other businesses, because i’m fortunate to be able to do this full time now, and most of my life is here at the farm. Including high salinity, high ammonia concentrations and low oxygen levels.

aquaponic system blueprints
aquaponic system blueprints

This was what we did with the prawns, because we are so busy running the farm to worry about much else. In colder climates, a small greenhouse. Once your seeds are safely planted, you can turn on your grow light. 1 trillion in the first year and balance the budget in year 4 of his presidency. With a limited pocket of funds, i was fortunate to be given a 5000 liter tank. On average, plants can be grown at the planting density shown below. He said that he had checked in with the office manager that day and was told to be there by 7:00. You just saved a ton of money on building your farm by adapting an existing one. Also apply a thin bead of the sealant under the lip of the waterfall weir to adhere it to the spillway.

According to reinecke, obtaining an up-to-date count of the total number of animals in a fish facility is an essential task, which is performed by technicians who manually extract animals with the help of small fish nets. One of her two dogs is dragged in with her. Another big question in designing an aquaponics system is the ratio of growing space to tank space. The main problem with peat is that it isn't. First 20% water is filled, then followed by the remaining after 2 days. In an aquaponic system, the used water from the aquaculture sector, which is rich in nutrients, is used to grow the plants. See the links below for top farmer’s free instructions on how to build a small-scale system at home. In such a small ecosystem ("mesocosm". Com/ visits the aquaponic place in waimanalo, hawaii to share with you what exactly is aquaponics, and how it works.

Food - beer - raffle - live music - donationsÂ  6pm â midnight friday aug. •educators – if you are an educator, you will quickly learn and understand that aquaponics is a great hands-on learning tool to teach all facets of science, agriculture, engineering, mathematics and much more. Then, add half a cup of baking soda. I deff wanted to try those but thought they would not do well in aquaponics cause thats what the overton fishery people told me lol. And sustainable systems based almost entirely on thermodynamic considerations.

Best fishing trip i've ever had or ever will have. Gruen noted that evo farm’s ability to accommodate groups as well as rosenstein’s facility in explaining how aquaponic systems work made her want to bring him in as a specialty teacher for venice high school students. So how can hydroponic and aquaponic systems reduce water loss, increase water use efficiency, and use water more sustainably. The downside of placing your pond in direct sunlight it may be uncomfortable for you if you want to sit by your pond and enjoy the fish. Instead, there are specific formulas to account for the reduced flow along the bottom and shallow sides of a stream. Incidentally, freshwater clams can make “pearls” and they command a high price. Aeroponics is the most unique hydroponic system.

They are then put into a spinning chamber and are spun together into fibers, much like cotton candy. Yes, mkristovskis, that is too small for 2 comets. The largest i've harvested so far have been 10 inches total length which are about 9 ounces. New information occurs daily in the world of aquaponics. In the city of manchester in the uk, unoccupied flat roofs account for an area of 136 hectares, representing one-third of the city’s inner urban area. Aquaponics brings the best of these two processes disposed of in some manner in order to keep the tank water free. Strapping on the back side of the two 'heavier side' magnets. A quick search for aquaponic farms returns sites with how-to aquaponics tips, pictures of homemade systems and small-scale farms. Let’s go over how to care for a plant.

 no need to battle early spring weeds or any of the other headaches typically associated with working the land after a hard winter. Braseth learn to build your own hydroponics system it s a method. It’s cute; it’s crafty, it’s spicy-and easy to put together. And i can’t throw enough food at them. The smallest to date was a 3 x 3 m balcony. A wicking bed is even more like a traditional raised bed because it actually has soil.

Penn plax aquaponic fish tank. The columns should be date, time, water temperature, ambient temperature,  ph, nitrate, nitrite, ammonia, do, yield in lbs, comments(includes what and when plants are introduced  and harvested, problems, any additions to systems) . (to be clear, they might be able to afford land, but at such a cost as to risk the enterprise and family. Though he is concentrating on commercial aspects, masabni said the idea is also inspiring people to try aquaponics on a smaller scale for personal consumption. For the recirculation system i think i got it now i'll use a koi pond deluxe aeration kit and. The fresh food and that because of all the food and it also helps us to aquaponics system: look at the fish tank could ultimately determine how fast your having a lot of poor information as you set your heater before the water around fish and plants. This was probably the last straw for the previous occupants; there were other problems with the house, but having to go across the street to use the neighbor's bathroom is pretty final. Historians have traced the use of siphons back to ancient egypt.

The main signal that tells a marijuana seed to start sprouting is the presence of moisture and heat. On billboards magazines and telescope eyes. “project goes with the flow…to haiti”-article featuring our work in aquaponics in haiti. Think of it this way:. I do grow those but they are the first to bolt to seed at the first sign of hot weather. Com/projects/sustainable-humane-earth… just 17 hrs left to contribute.

As charlie, max, and evan explained, aquaponics uses 90 percent less water than traditional farming, and the usda recently declared it can be an organic way of farming, so long as organic practices (i. Basically, anyone who is interested in building with wood can learn it successfully with the help of free woodworking plans which are found on the net. Blueprints for aquaponics systems are fairly rare on the internet. If it’s important to generate a paper of distinctive flawlessness, just order an essay listed right here and our writers can offer guide. Hydroponics involves the addition of chemical "fertilisers" and nutrients to the water, because plants could not simply grow in water alone- they need nitrogen mostly. I appreciate that your statements in this blog imply that are looking internally and asking “what is my piece, my responsibility for the situation we find ourselves in.

Expenses associated with raising our fall chickens. The benefits of this system are that it uses a lot less water, supports your root vegetables, and is flexible because you can grow multiple plants at once. For myself i find the videos most valuable to visually see what others have done and pick up little tricks. They have no moving parts and i love them for that. Not all are suitable, but there’s a few key species that are especially excellent for home aquaponics setups… .  i often take cuttings back to the house while out running and hiking. Cycling using fish can be stressful to the fish and will require at least a month before your system is fully functional.

Aquaponic System Blueprints

If you have a piece of plastic, you can drill holes lots of holes in it for air, that could work temporarily until you get a new habitat. This is my home and my hope is also that some of the fisherman and displaced businesses will find their way into some new opportunities such as this one, because .  so, for example, for every one pound of fish that i grow, i’m going to get 10-12 pounds of vegetables. Maintenance is simple, you just feed the fish and occasionally add some extra nutrients, then wait and harvest your crops. Aquaponics - pure, fresh, 100% organic, the future of nature protective farming.

I try to eliminate mechanical complexity and increase biological diversity. Types of bacteria are harmful. Here’s the link to that website by the way, hope this help people out there who are looking for livelihood training and/or additional capital for their business. The aquaponics system uses only two% of the water regular gardening does… since the water is continuously re-circulated , and you do not drop any to soil absorption. Firstly we’ll take a look at his ibc and blue barrel system in. Figure 5 shows the root systems for the growth of bacteria. Backyard aquaponics and practical aquaponics recommend a 1:1 or better a 2:1 ratio of growbed to fish tank volume. It is the maximal size of tank the manufacturers say you can use their filter. Next i think all the drain pipes should be covered with a thin layer of volcanic 3/4 gravel.   aquaponics systems use 95% less water than conventional farming, and it seems our whole world is in a drought and needs to conserve water.

Aquaponics used to grow marijuana in former pulp mill in nova scotia. You could give them meal worms or crickets, if you have them. Paul taylor has been an organic farmer for more than 30 years and has extensive experience bringing non-productive soils back to life, turning them into profitable systems. We harvest it not only through the summer and fall, but leave it in the garden through the winter. Many water gardeners may cringe at the. As a result of b.

Gallons per hour (gph) requirements for spillway width can be easily figured out with the following formula. One of the best technical articles online to understand the technology of aquaponics is “optimization of backyard aquaponic systems. My favorite part of focusing my efforts and concerns on food production, is that it makes me happy. Fast, in a system that feeds and waters itself. — which in aquaponics is utilized by the vegetation. The nitrogen cycle starts when fish feed is first added to the aquaponic system. Like other herbs, thyme could also thrive well when exposed in an aquaponics media bed. Step 9: cycling your aquaponics system. Heirloom seeds versus hybrid seeds:.

  we will update it to include imperial measures to save our international readers which use this system from converting the formulas to suit. M a r k e t s, o n t h e ir. So, i'm not able to use this pump type in-line, but i will be using it in the winter in the fish tank once i shut down the grow bed. It is geared toward people who want to build their own aquaponics system but don't know where to begin. Additionally, pressure compensated emitters must be used to ensure that each pot receives the exact same volume of irrigation solution from one end of the bench to another.

Aquaponics system vegetation and if your rising edible fish, then additionally they present protein for yourself. Simply follow the steps, and design the system that you want. Hydroponics, aquaponics, and container growers cheered the decision. Some of these tools may be a water chiller, water heater preferably thermostatically controlled and a digital thermometer.  potential partnership with similar growers, research labs, government bodies. Storage: store picked tomatoes at room temperature indoors, or in a shady place outside.  castings improve soil by supporting the soil’s living systems, what’s known as the “soil food web”. Wasabi has an anti-bacterial agent help to neutralise any bacteria in the food. This returned water would mix with oxygen in the atmosphere, increasing the overall oxygen level in your aquaponics system.

Usage: this work is licensed under the creative commons attribution-noncommercial-no derivative works.   the other system involves the construction of. The growing beds can be placed at waist height so there's no need to bend over. There are three families of crayfish, two in the northern hemisphere and one in the southern hemisphere. This is where the beneficial bacteria turn the fish waste into usable, nutrient-rich food for the plants. For example, the use of total exclusion netting and other ‘soft’ (non-toxic) methods of pest control and management may be viable with the current invention that may be not cost effective for less intensive greenhouse systems. One of these will be the site of the first greenhouse and at the rear, northern edge i'll build a living quarters with a green roof. Tend not to miss get particular offer for aquaponics4you is a digital plan (which includes video guidelines) that shows you how to develop your extremely own aquaponic program at home.

Many of the system designs available from web sites at present are designed using two pumps or oversized pumps for the application required, which is completely unnecessary and energy inefficient. Here at grow dinner aquaponics we raise freshwater prawns and tilapia in our aquaponics. In order to create presentable crops and meat the dna is altered to insure the size and growth rate of produce are both far larger, and that crops are resistant to specific pests and disease. The grapefruit is a subtropical citrus tree, and the fruit is well-known for its semi-sweet to sour taste. Springwork’s hopes to have the new greenhouse operating by the summer of 2018. Aquaponics system blueprints that helped me. I've been keeping the running bamboo small and in containers here as the climate and soil could spell this bamboo taking over the neighborhood if i don't contain it.  he just looks for solutions at a molecular level. And, in new orleans, marianne cufone uses her in-ground aquaponics system—in which ponds full of fish provide nutrients for the trays of herbs, cucumbers, and melons floating above them, and vice versa—partly as an educational tool. It gets hot in the summer.

Like owning a dog, there is a reason they are not for beginners. Aquaponics systems can provide an environment of sustainability, particularly in “backyard aquaponics” type systems. Guess which free, bud lite-hued high-nitrogen fertilizer i’d recommend.   read up on nitrifying bacteria until you understand and can predict the cycle. Working with our members to foster the sustainable growth and development of the uk aquaponic sector. Only the aquaculture components of the uvi system are covered by bird netting. It is a huge amount.

He was in china and hong kong specifically for a while, now at the border between columbia and venezuela. It is creativity ranked fifth from leaping out;. A sponge filter will act as a mechanical and biological filter, and the bubbles will keep the water moving. Rooftop chickens might be another, balancing, system, giving, eggs, occasional meat and of course fertiliser to feed those plants you do grow in beds (root crops and brassicas, although even these. In aquaponics systems, the roots of the vegetables will mostly be immersed in water instead of soil, so this system isn’t a good choice for root crops. As i and others have pointed out before, your edit is seriously flawed in multiple respects, which makes your edit unacceptable as a wikipedia article. Inside the ebook you will discover every thing you require to make your own aquaponics method, including diagrams, explanations and directions, each-thing is step-by-step, anybody can do this. The recycling of water takes care of the twin issues of waste water disposal and sourcing of fresh water.

Aquaponics 4 you is easy and safe to download. So if you are having trouble finding any aquaponics system blueprints then i recommend you head over to aquaponics 4 u and consider what they offer. Because water is recycled through an aquaponic system, only a small amount is needed to replace waters used through vegetable growth and evaporation. The best time to fertilize is when you change the water.  glue the two sheets together. The concept behind the triple bottom line is that companies are responsible first and foremost to all their stakeholders, and these include everyone that is involved with the company whether directly or indirectly, as well as the planet we're all living on. If we take the example of the single ibc tote ap system, it will operate with the pump in the fish tank and will fill up the grow bed, same as most domestic ap installation.

This isn’t to say that people should avoid meats or dairy, or yummy desserts for that matter, but worms love organic food scraps so it’s always good to have a little on hand, and most people do. What works well in deep water culture troughs are kale and chard and other leafy greens like the different varieties of lettuce. This company sells landscaping supplies. Now if you have a 10-gallon tank, this is a viable aquaponics option. So, once ammonia is in the solution, it must be transformed or it will eventually kill the fish.  since aquaponics is primarily focused on terrestrial plant production, we're going to keep it simple and restrict this discussion to terrestrial plants and nitrates.

This was the same issue with the bead filter. I have gardened in montana for over 60 years and am a big fan of food storage. The nop (national organic program) and nosb worked to insure animals have access to pasture and a connection to the soil. Help and be used by aquaculture businesses, aquaponics culture, and the aquarium industry. Perlite is a common, soil and gardening amendment you can find at your local nursery or hardware store. Some systems have grown fruit trees aquaponically, but this is still experimental. [blockquote cite=”tom losordo” type=”left”]don’t be impressed with high fish densities. Put the ball valve on that flow and adjust the ball valve till the flows are balanced as needed.

The fruits seem to be larger and the growth is more consistent and they produce better. If you’re building up from the middle of the tank (often the easiest and most attractive way of doing it), make sure that your structure is at least two thirds the height of your tank, and wide enough at the bottom not to look “tippy” or unstable. This system will become a hurricane force low. Many people say aquaponics is organically certifiable. The microbes are a commonly overlooked part of an aquaponic system, but it is these that do the most important work in the nutrient cycle. I'm working on a design for my new office, and i might just take you up on that offer. Backyard aquaponics kit within mobile phones, mobile computer or pc by simply right-clicking the actual picture after which select. You bet, and now that you know what urea is and how it can benefit your garden, are you more willing to experiment. Aquaponics is a new type of farming which relays on the symbiosis between the plants and the fish. If the water does not meet these criteria, corrective actions are required as soon as is practicable, but no later than the following year.

They don’t like it when guys calculate their weight. This day we tried a new item, the thai coconut meatballs.   this can be a very cost effective way to. Permaculture simply refers to a garden that can essentially take care of itself. The next year, we brought our little puppy home, an irish wolfhound. In this article, we will understand how these systems work, some applications and designs of all three systems. The understandings, to together achieve the goal of perfection.

With many people looking for ways to increase their income these days, starting a home-based business is an attractive option. The stiff staghorn’s fertile leaves are narrower than the shields, but are still wider than other platycerium varieties with shallow lobes as well. I have recently pulled two potato bushes out and was a little disappointed with the harvest from them. When dissolved oxygen is sufficiently present the iron and oxygen bond together. When you purchase a pump, the gallons per hour (gph) is based on a head height of zero.

The aquaponics plans they provide are not exactly like blueprints but more like a step by step process to making a aquaponics system. Aquafarm aquaponics fish garden australia. It breaks apart very easily. Contact us to get more information on how to get started with aquaculture or to find out about our aquaculture systems and tanks for sale. The exporters pay you at time of pick up at a rate dependent on time of year of $11 up to $15/kg. Are you going to be able to take the ship in from here. Diy projects are fun and often save you money, but not all of these types of experiments are successful and practical. Once you have built or assembled your system, the first few days and weeks will require very close supervision to make sure everything is working and to encourage the system to achieve a good equilibrium. The aquaponics system for those in the market today are presuming that you want to understanding of the cripple creek colorado.

In a well built system like ours, flood and drain aquaponics has proven to be durable and robust. The idea is you flood the growbed with the fish water (delivering nutrients and, um, water), then let it drain out (bathing the roots in air/oxygen). Be forewarned that since all of them live together, outbreak of disease is very detrimental and it could bring disastrous effect. Have noticed that there is a lot less leaf damage on my plants this year. In fact, we have made our focus vertical aquaponics.

When you grow your plants indoors or in a confined space, you want to ensure they are getting the best nutrients to their roots, more than enough air and water, are allowed maximum room for their root growth and can be physically supported as they grow. Coming back home with a wealth of information. Compeople always wonder, “what are the best plants for aquaponics. It was unbelievable, that first juicy bite into our aquaponic tomatoes. All aquaponics experts agree: the most important thing to be mastered in aquaponics is the “nitrogen cycle. Provides a renewable resource that can be shared or traded. I was shocked, betrayed, hurt, dead ####ing broke, ashamed of my gullibility, and totally peeed. Another option would be to support the bag so it is only part way in the system water for 15 minutes (do not do this in the sun. I will have to look into this further.

No tilling the ground and incredibly little weeding.

Aquaponic System Blueprints
Permits are good for one year and come with detailed instructions on recertification. Super costa...


Boys fishing in Corpus Christi

Fishing after the holidays on the Nueces River

Boys Fishing in Corpus Christi

SONY f/7.1, 1/640 sec, ISO-400

For the holidays, the family and I went down to the Corpus Christi area, where we visited my family that lives in the area. While there, the boys wanted to go fishing. My dad and youngest brother, Julian, go fishing a lot, so they wanted to join them one afternoon. On an overcast day, we traveled down the Joe Fulton corridor to find a small park along the Nueces River.

Boys fishing in Corpus Christi

Since the boys have never been fishing, this was a new experience for them. As you can see in this photo above of how my youngest son, Manu, was holding his fishing rod as he reeled in nothing on the line. This would be his last time because his attention span was taken over by his surroundings and went to go play with the sand instead.

Sand throwing in Corpus Christi

My oldest son, Jay, had been the one adamantly asking about fishing and was really trying to reel some fish in. There was nothing on the line, but he waited for about 30 minutes before he realized how much fun his brother was having playing with the sand and abandoned his fishing pole.

In the above photo, you can see Manu and Julian tossing the sand while they leave their fishing poles in the ground, waiting for a bite on the line. The images I tried to capture were of the calm skies and trying to recreate those old fishing days I remember from when I was their age. The overcast day allowed me to keep the natural colors, and I sparsely had to change my camera settings.

Dad fishing in Corpus Christi

Fish in a bucket

SONY f/7.1, 1/400 sec, ISO-400

The only difference is my Dad in the first photo, is now older than when we first started going fishing together back in the day. So the fish weren’t biting at all. Maybe it was the time of day, or perhaps the fish were away for the holidays. Fishing is a patient sport and growing up near the ocean, it’s one of the best ways to pass the time. We do know there were fish there. Another fisherman near us caught a medium-sized catfish.

Before we arrived, they had already caught some small fish to use as bait after throwing a large net near their fishing spot. I was working on some close-up photography, so I thought I’d share an image of the only fish we saw that day.

We spent about an hour out there with the boys. They were starting to become grouchier without a nap and throw sand in the water, scaring whatever fish that may have been near us. Ask the boys about the fishing trip, and they’ll tell you about all the fish they caught. Fishing teaches you how to tell tall tales.

Boys fishing in Corpus Christi

Triceratops Skull

Dinosaurs down in Seymour!

Dimetrodon Sculpture at Whiteside Museum

SONY f/4.5, 1/1000 sec, ISO-400

This past Saturday, the boys and I traveled down to Seymour, TX, to visit the Whiteside Museum of Natural History or as it’s best known around here as the dinosaur museum.

Seymour is a town that has almost 2000 people in it. This place is true small-town Texas. So when you tell people there is a dinosaur museum down here, it really baffles them. The closest big city, Wichita Falls, is about 40 minutes away.

Once you realize this is literally the middle of nowhere, you come to a gravel parking lot with huge animals painted on the side of a building. This is the Whiteside Museum and home to the North Texas dinosaurs. The museum features animals in this area, including the dinosaurs that once roamed this area millions of years ago. You’re greeted by a Dimetrodon sculpture, the local dinosaur that made this place famous for paleontology digs. Don’t forget to take a picture with it, because it was made to be shared on social media.

Dimetrodon exhibit

SONY f/2.5, 1/400 sec, ISO-5000

Most museums I’ve seen with dinosaurs have favorites like the Stegosaurus or Tyrannosaurus Rex fossils. This museum had only the locals, which was surprising to me, that at one time, Tyrannosaurs Rex did roam this area. There is a giant skull located inside the exhibit as well as a replica head of how big they would have been.

Tyrannosaurus Rex Eye

SONY f/2.5, 1/640 sec, ISO-5000

I think what most impressed me was how the exhibits were set up. Everything looked state-of-the-art without being state-of-the-art. Small lighting tricks and great use of space make this place look bigger than it is. This doesn’t take away from the experience either. The amount of fossils available to you to view is astounding. Some stuff you’d only expect to see in a major museum, but here they were right in front of you.

There are also models of the dinosaurs, with many having hair to emphasize the newer interpretations of these giants beasts. Personally, my favorite was looking at the Triceratops skull.

Tyrannosaurus Rex and other dinosaurs

The place is great for kids too. Both of my boys enjoyed the experience. My oldest son is really into dinosaurs right now, so he was in a trance the whole time he was there.

We went on a weekend, and there were a couple of places that looked closed, such as the area where they clean up after the digs. There is glass around there, so you can watch them working, but no one was there on a Saturday. They’re also working on a new exhibit that will be about skulls.

Triceratops Skull

SONY f/1.8, 1/250 sec, ISO-5000

Aside from the dinosaurs, they also have local animals like iguanas, horned lizards, and turtles in exhibits to see what real-life reptiles look like. Walking into another exhibition are animals from Texas. You start at the front of the museum with the dinosaurs and then travel through time to the current animals that are alive in our time.

If you do go, you can see everything in about an hour. There are admission prices, so keep that in mind as well. There is also a gift shop in the front that has a lot of hard-to-find dinosaurs. Overall, it was fun to learn about dinosaurs that roamed around the North Texas area, the boys, and I call home.

Triceratops Skull

SONY f/2.5, 1/250 sec, ISO-5000

Aztec Dancer

Day of the Dead in the Falls!

Dia De Los Muertos Ofrenda

SONY f/5.6, 1/125 sec, ISO-20000

I didn’t have to travel to Mexico to celebrate Dia de los Muertos. While that is still on my bucket list, I was able to enjoy it in my backyard right here in downtown Wichita Falls. It was a fun celebration all-around, and I’m happy to see it continue growing in its third year here in Wichita Falls.

Fire Twirler

SONY f/5.6, 1/400. sec, ISO-16000

The latest lens in my photography arsenal is the Sony 100-400mm f/4.5-5.6, which I use when I need to be farther away. In this case, for the parade and the small spaces, it came in handy. Combined with the Sony A7riii, it allows me to photograph in low-light. The picture above allowed me to be far enough away, but still snag the action.

Parade Spirit

SONY f/5.6, 1/400 sec, ISO-16000

Aside from the parade, there were ofrendas set up in the middle of the festival. Ofrendas are offerings that are made to our ancestors during Day of the Dead. We place a photo on the ofrenda, hoping they come back and spend time with the family celebrating their life. It’s a way for us to remember our loved ones and what made them special. The food we offer is usually stuff they enjoyed when they were living with us.

In the photo from the beginning, you can see how beautifully these ofrendas are made. With lights for the night and marigold flowers to guide the spirits back to their family.

SONY f/5.6, 1/125 sec, ISO-8000

You would also find Catrins and Catrinas all around the festival. The one above was in the ofrenda area. People will also paint their faces as a Catrin to celebrate loved ones. Yes, in Hispanic cultures, skulls are a big part of the Day of the Dead festivities.

There were big Catrins for people to take pictures with. All around were booths selling food and merchandise. Music played on the big stage for the night. In the middle, in front of the ofrenda area, a circle had formed to see all the performances scheduled.

Aztec Dance

SONY f/7.1, 1/400 sec, ISO-4000

Something that the festival had last year was the Aztec Dancers. They recently were in the Dallas Day of the Dead celebration and then made their way to Wichita Falls, where they performed last year.

They did a blessing for ofrendas and also did the celebratory dances that span 200 years of family generations. They originate from the heart of Mexico in Mexico City. It was fun watching the dances and the colorful costumes. It gave authority to the festival and made it feel like we had gone to Mexico if only for a night.

Aztec Dancer

London Underground

What I saw overseas!

Keble College

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I’ve been back in the states for about a month, and I’m still pouring through thousands of photographs. See all the photos from the United Kingdom makes me want to go back, if only for a weekend.

One place that was probably one of the most gorgeous places I visited in the UK was Keble College in Oxford. This place has that college look you always think about when going overseas. It has the old traditional-architecture and the stained-glass in every building. The grass is also immaculate, and you are discouraged from walking on it.

I was able to attend Keble with the business class, which takes a trip to Oxford to learn about the Mini-Cooper plant. Because it’s a long day, we stay in Oxford and use the college as a place to sleep for the night. Everything about Keble reminded me of the Harry Potter movies. Ironically, the college was used as inspiration for the film. You can see that in the Great Hall photograph below where we were able to enjoy an excellent breakfast buffet. The bacon they have in the UK is tasty.

Keble College Great Hall

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Just sitting in the Great Hall, I felt like Dumbledore was about to come and give me points for just waking up early. The students enjoyed it and were snapping pictures everywhere. You could tell we were tourists because we wouldn’t stop snapping pictures in what felt like a sacred building. It was just a cafeteria with great architecture and servers. I wouldn’t mind going back to Keble and photograph more of Oxford.

Millennium Bridge

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Back in London, we saw more sights, including the Millennium Bridge. There were quite a few photographers also on the bridge. It was hard to show the right shape of the bridge while on it, so I had to wait until I was quite a bit away to take a decent photograph. My favorite part was being able to show the older St. Paul’s Cathedral in the background with the new futuristic Millennium Bridge. For my Harry Potter fans. This bridge is in the first part of the movie Half-Blood Prince.

London Pride Parade

The first Saturday of my time in London, we took a trip to Trafalgar Square. We were just passing through, but I, along with some students, found ourselves lost in the crowd. I was busy taking pictures, and the students were just enjoying the festivities. I tried to capture how big the groups of people were, but there was just no way with how packed it was. I went to Trafalgar Square a few times after that Saturday, and I never saw it packed liked it was for the London Pride Parade. With crowds everywhere, I tried to focus on couples and then let everything in the background fade out.

London Southbank Stake Space

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The South Bank Skate Park was full of creativity. From the skateboarders learning new tricks to the graffiti art, to the small market. There was something for everyone, and colors were popping everywhere. Each graffiti piece was so detailed, I tried to focus on the more significant parts. The girl in the painting above had a striking pose and was in just enough sunlight to give me some good exposure. It was a cloudy day, so I did my best to adjust to the ever-changing sun patterns.

Piccadilly Circus

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In the Piccadilly Circus, we saw street performers and crowds all over. I just tried to concentrate on emotion while taking these photos. The architecture was great, but when there are so many crowds of people. You just try and photograph emotions to tell the story of the area and what people are doing there.

Globe Theatre Balcony

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I posted about the Globe Theatre and my experience sitting in what would have been in their time, the luxury boxes. This was at eye level before the performance began. It gives you a better view of what I saw for most of the night. The detail in this place is impressive. I did some highlight compensation in Lightroom to try and bring out the lighting in the statues on the orchestra balcony. Doing this, I had to put the color back in the hanging pillars of fabric that made the show so colorful.

If you ever have the chance to visit London. Make sure to stop by and catch a show here. You won’t regret it and don’t forget to sit in the middle sections to experience the show from a new point-of-view.

London Underground

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I’m wrapping up my time from London in this blog, but I hope to go again. I fit a lot of adventures with photography into two-weeks. Until next time. Cheerio!

A couple of nights in Paris…

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I’m back in the states, but I still have a lot of photographs I’m pouring through. On a free weekend during my trip to the United Kingdom, I decided to make a trip to Paris, France. Unlike living in Texas, everything is only a few hours away in Europe. Two and a half hours on a train and going underwater through a dark tunnel was all I needed to find myself in Paris.

I always dreamed about taking photos of this place, but never did I think I’d actually be able too. I had to force myself to just enjoy the sights and not look at everything through the lens of my camera. As a photographer in a new place. The first thing you want to do is photograph everything. Sometimes it’s best to just take it all in the first time.

The photo above was from my first night there and exploring the outside of the Louvre Museum. The Louvre, of course, is the most famous museum in the world. At night the glass pyramid that rests in the middle of this once fortress lights up in the Paris night.

That night, there was a concert, so I also was able to photograph rows of people watching the musicians performing.

Paris Streets

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The Louvre lights up everywhere at night, and you don’t even need a tripod majority of the time. A tripod will help, but there is so much light around the area, just adjust your settings accordingly and be still while you photograph the city. You’ll see some excellent photographs come out of your camera, like the one I took above, of a small restaurant. There are small restaurants everywhere, so enjoy the culinary experience that is Paris. I could do a blog on just the food I tasted.

The lighting at night at the Louvre and everywhere in Paris is spectacular. During the summer, the blue skies stretched out into the late night, and I still saw brilliant skies well past 10 p.m. It’s more like a golden two to three hour in the city of lights.

Paris Streets

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While I could have stayed out all night, I did need to get some rest before starting my journey to actually take in the sights of Paris. The incredible architecture photograph above is one example of the buildings found around the city. This was near their art district.

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The Louvre was surrounded with statues inside, outside, and even sitting on the ledge. Since they didn’t allow visitors to go on the balcony, I had to take this one from a window and do some editing to make the water spots on the window disappear. This statue just looks down and stares at the hoards of people that visit this museum every hour of the day.

Hanging out near those windows on the top floors gave me an elegant look around the Louvre. The area used to be a fortress back in the day, but over time it has become a mecca of art.

I know I’ve mentioned the Louvre a lot, but I did spend most of my time there. I even had a hotel nearby called the Hotel Prince Albert Louvre.

Hotel Key

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This is a quick reminder that a lot of places in Paris, as well as Europe, do not have air-conditioning units in their rooms. After a week in Europe, I was already used to not having it. All I did was open the big windows and let the air circulate. The hotel was quaint and resembled something from the 1900s. My favorite part was the key in the photo above. No card key at this hotel. It gave me that old-world feeling.

Next on my list was the iconic Eiffel Tower. I was only there for a couple of days, so I did the most touristy things I could.

Locks at the Eiffel Tower

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The locks were everywhere, and not just near the Eiffel Tower. I initially couldn’t find many of them on the bridges, but I did find them everywhere else. I snuck in this photo on stairs that led to the bottom of a deck before another couple obscured my view by placing another lock on the railing.

Eiffel Tower

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The Eiffel Tower is large and impressive, but I believe the view is better faraway. This is a personal preference, but I was able to do more with the Eiffel Tower and skies when I was farther away. I lined up the Eiffel Tower and managed this shot as the sun was going down. If you are photographing the Eiffel Tower, remember that at night it has to be pitch-black before the lights start to sparkle on it. It really doesn’t feel like Paris until you do visit the Eiffel Tower.

If you ever have a chance to visit this city, remember to take a bunch of camera cards. You’ll have an excellent photograph on every corner.

Lion and London Eye

Found my self overseas!

London Blue Whale

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Visiting London wrapped up my summer of study abroad doing marketing, social media, and photography. London was different than Grenada but also different from the living in the states.

While I was there, I covered students attending classes. There was nine classes total, and I almost made it to everyone. I was only there for seven teaching days, making it a challenge to see every class. The above photo was from seeing the Natural History Museum in London with the education class. The Blue Whale skeleton, named Hope, hung in the middle of the exiting gallery leaving the museum. Trying to line up photographs like this take a few shots, but I measured lines on the ground and kept looking at the side angles to make sure it was aligned as symmetrical as I could make it.

Queen Mary University of London

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I spent two weeks in London with the study abroad group, and we stayed at the Queen Mary University of London. The school regularly has students, but since it was the summer, we didn’t see a lot of college students. There were plenty of other school groups there, including us with over 100 students. One of the largest classes from Midwestern State University.

I have so many pictures from the trip, I’ll split this up into two blogs, including some of my favorites.

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One day I was with the history and sociology class on a boat, punting along the river. The next day, I would be at the Camden Markets looking at the local shops. Punting was in Cambridge, where we spent half the day learning about the town and seeing the other colleges in the area. Punting means pushing, and Punters would sit on the bow of the boat, gondola style, and punt down the river while giving us information about the river and area.

Camden Markets

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The Camden Markets were located in north London, and it was a bunch of shops all put together. The one in the photograph was an Indian shop selling stuff from India. The lamps struck at me and created this pattern throughout the shop. Of course, I stuck a while until I photographed the image I wanted. At the markets, I found an art print to include in my collection. The markets also had a selection of food to eat from around the world.

Globe Theatre

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One of my last nights in London, I went with the Shakespeare class and saw a performance of “A Midsummer Night’s Dream” at the world-famous Globe Theatre. The students already had tickets, so I was stuck with the expensive ones, which were really not that expensive. I did have an enjoyable seat, as you can see from my angle. The students stood for the entire play, to have a real-life look at how it would have been in the times Shakespeare lived.

Some of them even participated in the play by doing minor-tasks. The production was a lot of fun with great acting, music, and incredible dance numbers. This was the way Shakespeare was meant to be performed, and you can see it at the Globe Theatre in London.

The other big thing about London is how you travel. From the buses to the tube to an actual car, there are many ways to see the city. The ‘Mind The Gap’ painted on the picture below was a constant reminder every time we used the tube.

Mind The Gap

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It was a warning to remind passengers to be careful as they left the tube. If there is one thing you will remember about the tube, it’s that little phrase.

Lion and London Eye

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Here is a photograph of a Lion with the London Eye in the background on a cloudy day. There were a lot of cloudy days, and the weather was always around the 70s. Apparently, this is hot, according to the locals. There was also rarely air-conditioning around, except for a few places. It’s manageable, and I became used to it after a few days in London.

London was a lot of fun and has some of the most exciting sights in the world. It was great to have a chance to visit and photograph this great city that plays host to the capital of the United Kingdom. Stay tuned for more from London and also my weekend in Paris.

Art Battle Wichita Falls

The first rule of Art Battle, is we don’t talk about Art Battle in Wichita Falls!

Art Battle Wichita Falls

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The Wichita Falls Art Association is celebrating its 70th anniversary. For their anniversary, they decided to bring Art Battle to Wichita Falls for the first time. This event was so much fun and showed how much talent Wichita Falls has in this town.

It also doubled as a fundraiser for the art association, which included a silent auction of the art that was being created right in front of everyone. It was held inside The Warehouse, a newer venue of a remodeled warehouse in downtown Wichita Falls. It really gave the impression of the movie Fight Club.

For the event, my wife, Teresa Lopez, was one of the competitors in Art Battle. She was selected along with 11 other artists. There were two rounds of six artists, and each artist has 20 minutes to paint a piece. Artists also have to use a specific paint palette chosen before the time begins and cannot have any reference materials. The top two artists from each round made it to the final round.

Art Battle Wichita Falls

After the first round, Teresa was one of two artists to move on, making it in for her art piece of a person performing martial art on top of a mountain with a jellyfish in the starry sky.

Art Battle Wichita Falls

In the final round, she made a sunset with a starry sky and a rabbit with long ears sitting on top of a mountain. Teresa didn’t win the whole Art Battle, but she had a lot of fun and was able to hang out with some old artist friends from college.

I was able to play public relations for her and take photos of the Art Battle and center only on her. This is a change from my day job, so it was nice to just focus on one subject. My camera settings didn’t change much. I focused on where the spotlights were shining, and once I had my settings adjusted for them, I didn’t really stray away from the lighting. In post-processing, I played with the shadows to bring back the light into the photo.

The Art Battle is coming back in August with more artists, and I’m glad that Wichita Falls has another fun event to be a part of. Go check out Art Battle Wichita Falls.

Art Battle Wichita Falls


Author Archives: chrisharrow

Inscribed Triangle Challenge

@MathCeyhun posed an interesting geometry problem yesterday.

Even more interesting is that, as @MathCeyhun noted in a subsequent tweet, none of the posted solutions is correct.  There have been a few posted answers, but no solutions, so I thought I’d give it a try.

  • The perpendicular bisector of each chord was given, and perpendicular bisectors of chords always lie on radii of the circle.
  • If r is the radius of the circle, then the lengths of the extensions of the perpendicular bisectors are r-1, r-2, and r-3.
  • Nothing given guarantees anything special about the triangle, so I assumed it was scalene.  I called the side lengths 2x, 2y, and 2z to simplify the bisection labels.
  • Adding the bisector extensions, the radii to the vertices, and segment names and labels gave me this.

[Irrelevant to this problem, but I just realized by looking at this image that every triangle can be dissected into three isosceles triangles with congruent sides and a shared vertex point at the triangle’s circumcenter.  Pretty.]

Each isosceles triangle is bisected by the perpendicular bisector of its base from which I extracted three relationships from the Pythagorean theorem.

x^2+(r-1)^2=r^2 \longrightarrow x^2=(2r-1) \cdot 1

y^2+(r-2)^2=r^2 \longrightarrow y^2=(2r-2) \cdot 2

z^2+(r-3)^2=r^2 \longrightarrow z^2=(2r-3) \cdot 3

[The relationship between the half-sides, the extension of the radius beyond the triangle, and the radius is another pretty pattern.]

That gives 3 equations in 4 variables.  I needed one more to solve….

The area of \Delta ABC can be expressed two ways:  as the sum of the areas of the isosceles triangles, and using Heron’s formula.  From the areas of the isosceles triangles,

Area( \Delta ABC) = \frac{1}{2}(2x)(r-1) + \frac{1}{2}(2y)(r-2) + \frac{1}{2}(2z)(r-3)
Area( \Delta ABC) = x \cdot (r-1) + y \cdot (r-2) + z \cdot (r-3)

The sides of \Delta ABC are 2x, 2y, and 2z, so its semiperimeter is x+y+z and Heron’s formula gives its area as

Area( \Delta ABC) = \sqrt{(x+y+z)(-x+y+z)(x-y+z)(x+y-z)}.

The area of a given triangle is unique, so the two different area expressions are equivalent, giving a fourth equation.

With four equations in four variables, I had a system of equations.  The algebra was messy, so I invoked my CAS to crunch it for me.

The question asked for the area of the triangle, so I just substituted my values back into the area formulas.

And 17.186… is clearly not one of the choices in the original problem.

Recognizing the perpendicular bisectors, seeing all the right triangles, and connecting the multiple ways to describe the area of a triangle made this solution reasonably easy to find with the help of my computer algebra system (CAS), but I know the background algebra is, at best, cumbersome.  I hope there’s a more elegant solution, but I don’t see it.  Can anyone offer a suggestion?

Either way … this is definitely becoming a challenge problem for my Quantitative Reasoning class this coming week!

Implicit Derivative Question

Yesterday, this question posted within the AP Calculus Community about implicit derivatives.

Below, I argue why the derivatives MUST be the same, show how four different variations can all be shown to give the same derivative, and provide a final conclusion.

The Desmos graph of the given relation, is y^2 = \frac{x-1}{x+1} , is shown below.  Logically, it seems that even when the terms of the relation are algebraically rearranged, the graph should be invariant.   The other two forms mentioned in the Community post are on lines 2 and 3.  Lines 4, 5, and 6 show three other variations.  Here is the link to my Desmos graph allowing you to change between the forms to visually confirm the graphical invariance intuition.

If calculus “works”, it also shouldn’t matter how one calculates a derivative.   While the forms of the derivative certainly could LOOK different,  because any point on the invariant graph has the same tangent line no matter what the form of its equation, and the derivative of a relation at a point is the slope of that invariant tangent line, then the derivative also MUST be invariant.

To show the derivatives are fundamentally all the same (as suspected by the initial post), I calculate the derivatives of the equations on lines 1 and 3 given in the initial post as well as my variations on lines 4 and 6.

Using the Chain Rule on the left and the Quotient Rule on the right gives

This version is more complicated, requiring the Product Rule in addition to the earlier Chain and Quotient Rules.  In the penultimate line, I used the original equation to substitute for y^2 to transform the derivative into the same form as line 1.

This time, differentiation requires only the Chain and Product Rules.

After the usual substitution for y^2, I multiplied both sides by (x+1) to clear the denominator and solved for y', returning the same result.

This time, the relation is solved for x, resulting in a much more complicated Quotient+Chain Rule calculation, but substituting for y^2 and changing the form leads once again to the same answer.

Hopefully this is convincing evidence that all derivative forms can be shown to be equivalent.   If you’re still learning implicit differentiation, I encourage you to show the derivatives from the lines 2 and 5 variations are also equivalent.

So which approach is “best”?  In my opinion, it all depends on your personal comfort with algebraic manipulations.  Some prefer to just take a derivative from the given form of y^2 = \frac{x-1}{x+1}.  I avoid the more complicated quotient rule whenever I can, so the variation from line 4 would have been my approach.

The cool part is that it doesn’t matter what approach you use, so long as your algebraic manipulations are sound.  You don’t have to accept the form in which a problem is given; CHANGE IT to a form that works for you!

Pythagorean Investigation

Here’s a challenge @jamestanton tweeted yesterday:

is true?

Does anyone have a different approach?

Deep Unit Circle Understanding

Chris Bolognese started a new Honors Trigonometry unit a few days ago and posted a new problem set, welcoming feedback.  Clicking on the document image gives you access to Chris’ problem set.

This post is my feedback.  I wrote it as a conversation with Chris.  it

Question 2 suggests you have already discussed special angles (or your students recalled them from last year’s algebra class).  What I love about this problem set is how you shift the early focus away from these memorized special angles and onto the deep symmetry underlying the unit circle (and all of trigonometry).  Understanding deep structure grants always far more understanding than rote memorizations ever does!

The problem set’s initial exploration of trig symmetry starts in parts a-d of Question 1, asking students to justify statements like sin(x+\pi )=sin(x) and cos(2\pi -x)=cos(x).  The symmetry is nice, and I hope you refer back to this problem set when your class turns to its formal exploration of trig identities, helping them see that not all proofs of identities require algebraic justifications.

CONSIDERATION 1:  Put some creative power in your students’ hands.  Challenge them to discover additional algebraic/transformational identities like Questions 1a-1d.  For example, what is the relationship between sin \left( \frac{\pi}{2} + x \right) and sin \left( \frac{\pi}{2} - x \right) ?  There are LOTS of symmetry statements they could write.

CONSIDERATION 2:  I would shift Question 1e to Question 4, as 1e is the basis of the Pythagorean identities you explore in the later group.

CONSIDERATION 3:  I suggest dropping Question 4d and asking students to discover another equation showing another Pythagorean relationship between trig functions.  That cotangent and cosecant have not yet been used is hopefully a loud, silent hint.

CONSIDERATION 4:  They’re not ready for this one until they see sinusoidal graphs, but another trig pre-identity I love is graphing y=cos^2 (x) and asking students to write an equivalent equation using only translations and dilations.

Unlike the symmetric relationships in 1a-1d, I don’t think you can actually KNOW it is true without algebraic relationships.  I use problems like this to set up and justify the transition to identities.

I don’t recall ever seeing something like your Question 3 before.  In my opinion, trig1 is the gold nugget in the assignment, especially with your students’ prior exposure to special angles.

CONSIDERATION 5:  In some ways the large number of addends in trig1 gives away that there must be a simpler approach than the problem suggests on its surface.  What about something that suggests a direct solution if you don’t invoke the symmetry, like trig2?

CONSIDERATION 6:  Closely related to this, why not shamelessly take advantage of the memorized values to see if students notice the symmetry to notice a simpler approach?  I suggest trig3, with sine or cosine.

CONSIDERATION 7:  My prior two examples tweaked your initial trig1 problem.  Like Consideration 1, why not challenge your students to develop their own summations?  I bet they can develop some clever alternatives.

In Question 3, you had some nice explorations using degrees.  Unfortunately, I’m not aware of many equally clever early questions involving radians that aren’t degree-oriented in disguise.  Here are my final suggestions to address this gap.

CONSIDERATION 8:  Without using any technology, rank

sin(1), cos(2), tan(3), cot(4), sec(5), csc(6)

in ascending order.  Note that all angles are expressed in radian measures.  One pair of expressions is very difficult.  (A softer version of this question ranks only sin(1), cos(2), & tan(3).)

CONSIDERATION 9:  One pair of expressions in Consideration 8 is very difficult to rank without technology.  Which pair is this and why is it so difficult to rank?  Exchange angles or functions to change this question in a way that makes it easier to rank.

Thanks for the fun and thoughtful problem set.

FiveThirtyEight Riddler Express Solution

I’d not peeked at FiveThirtyEight’s The Riddler in a while when I saw Neema Salimi ‘s post about the June 22, 2018 Riddler Express   Neema argued his solution was accessible to Algebra 1 students–not always possible for FiveThirtyEight’s great logic puzzles–so I knew it was past time to return.

After my exploration, I’ve concluded this is DEFINITELY worth posing to any middle school learners (or others) in search of an interesting problem variation.

Following is my solution, three retrospective insights about the problem, a comparison to Neema’s solution, and a proposed alternative numeric approach I think many Algebra 1 students might actually attempt.

Here is a screenshot of the original problem posted on FiveThirtyEight (2nd item on this page).

If you’re familiar with rate problems from Algebra 1, this should strike you immediately as a “complexification” of D=R*T type problems.  (“Complexification” is what a former student from a couple years ago said happened to otherwise simple problems when I made them “more interesting.”)

My first thought simplified my later algebra and made for a much more dramatic ending!!  Since Michelle caught up with her boarding pass with 10 meters left on the walkway, I recognized those extra 10 meters as irrelevant, so I changed the initial problem to an equivalent question–from an initial 100 m to 90 m–having Michelle catch up with her boarding pass just as the belt was about to roll back under the floor!

Let W = the speed of the walkway in m/s.  Because Michelle’s boarding pass then traveled a distance of 90 m in W m/s, her boarding pass traveled a total \displaystyle \frac{90}{W} seconds.

If M = Michelle’s walking speed, then her total distance traveled is the initial 90 meters PLUS the distance she traveled in the additional 90 seconds after dropping her boarding pass.  Her speed at this time was (M-W)  m/s (subtracting W because she was moving against the walkway), so the additional distance she traveled was D = (M-W) \cdot 90, making her her total distance D_{Michelle} = 90 + 90(M-W).

Then Michelle realized she had dropped her boarding pass and turned to run back at (2M+W) m/s (adding to show moving with the walkway this time), and she had \displaystyle \frac{90}{W} - 90 seconds to catch it before it disappeared beneath the belt.  The subtraction is the time difference between losing the pass and realizing she lost it.  Substituting into D = R*T gives

\displaystyle 90 + 90(M-W)=(2M+W)* \left( \frac{90}{W} - 90 \right)

A little expansion and algebra cleanup …

\displaystyle 90 + 90M - 90W = 180 \frac{M}{W} - 180M + 90 - 90W

\displaystyle 90M = 180 \frac{M}{W} - 180M

\displaystyle 270M = 180 \frac{M}{W}

And multiplying by \displaystyle \frac{W}{270M} solves the problem:

\displaystyle W = \frac{2}{3}

Insight #1:  Solving a problem is always nice, but I was thinking all along that I pulled off my solution because I’m very comfortable with algebraic notation.  This is certainly NOT true of most Algebra 1 students.

Insight #2:  This made me wonder about the viability of a numeric solution to the problem–an approach many first-year algebra students attempt when frutstrated.

Insight #3:  In the very last solution step, Michelle’s rate, M, completely dropped out of the problem.  That means the solution to this problem is independent of Michelle’s walking speed.

Wondering if other terms might be superfluous, too, I generalized my initial algebraic solution further with A = the initial distance before Michelle dropped her boarding pass and B = the additional time Michelle walked before realizing she had dropped the pass.

\displaystyle A + B(M-W)=(2M+W)* \left( \frac{A}{W} - B \right)

And solving for gives \displaystyle W = \frac{2A}{3B}.

So, the solution does depend on the initial distance traveled and the time before Michelle turns around, and it was simplified in the initial statement with A=B=90.  That all made sense after a few quick thought experiments.  With one more variation you can show that the scale factor between her walking and jogging speed is relevant, but not her walking speed.  But now it was clear that in all cases, Michelle’s walking speed is irrelevant!

My initial conclusion matched Neema’s solution, but I really liked my separate discovery that the answer was completely independent of Michelle’s walking speed.  In my opinion, those cool insights are not at all intuitive.

While this approach is just a series of special cases of the generic approach, I suspect many Algebra 1 students would likely get frustrated quickly by the multiple variable and attempt

Ignoring everything above, but using the same variables for simplicity, perhaps the easiest start is to assume the walkway moves at W=1 m/s and Michelle’s walking speed is M=2 m/s.  That means her outward speed against the walkway is (2-1) = 1 m/s.  She drops the pass at 90 meters after 90 seconds.  So the pass will be back at the start after 90 seconds, the additional time that Michelle walks before realizing her loss.

I could imagine many students I’ve taught working from this through some sort of intelligent numeric guess-and-check, adjusting the values of M and W until landing at \displaystyle W=\frac{2}{3}.  The fractional value of W would slow them down, but many would get it this way.

I’m definitely pitching this question to my students in just a few weeks.  (Where did summer go?)  I’m deeply curious about how they’ll approach their solutions.  I”m convinced many will attempt–at least initially–playing with more comprehensible numbers.  Such approaches often give young learners the insights they need to handle algebra’s generalizations.

Painting and Probability

Here’s a cool probability problem the start of which is accessible to middle and high school students with careful reasoning.  It was posted by Alexander Bogomolny on Cut the Knot a week ago. I offer what I think is a cool extension of the problem following the initial credits.

The next day Mike Lawler tweeted another phenomenal video solution worked out with his boys during Family Math.

Mike’s videos show his boys thinking through simpler discrete cases of the problem and their eventual case-by-case solution to the nxnxn question.  The last video on the page gives a lovely alternative solution completely bypassing the case-by-case analyses.  This is also “Solution 1” on Alexander’s page.

When I first encountered the problem, I wanted to think about it before reading any solutions.  As with Mike’s boys, I was surprised early on when the probability for a 2x2x2 cube was \displaystyle \frac{1}{2} and the probability for a 3x3x3 cube was \displaystyle \frac{1}{3}.  That was far too pretty to be a coincidence.  My solution exactly mirrored the nxnxn case-by-case analysis in Mike’s videos:  the probability of rolling a painted red face up from a randomly selected smaller cube is \displaystyle \frac{1}{n}.

Surely something this simple and clean could be generalized.  Since a cube can be considered a “3-dimensional square”, I re-phrased Alexander’s question into two dimensions.  The trickier part was thinking what it would mean to “roll” a 2-dimensional shape.

The outside of an nxn square is painted red and is chopped into n^2 unit squares.  The latter are thoroughly mixed up and put into a bag.  One small square is withdrawn at random from the bag and spun on a flat surface.  What is the probability that the spinner stops with a red side facing you?

Shown below is a 4×4 square, but in all sizes of 2-dimensional squares, there are three possible types in the bag:  those with 2, 1, or 0 sides painted.

I solved my first variation case by case.  In any nxn square,

  • There are 4 corner squares with 2 sides painted.  The probability of picking one of those squares and then spinning a red side is \displaystyle \frac{4}{n^2} \cdot \frac{2}{4} = \frac{2}{n^2}.
  • There are 4(n-2) edge squares not in a corner with 1 side painted.  The probability of picking one of those squares and then spinning a red side is \displaystyle \frac{4(n-2)}{n^2} \cdot \frac{1}{4} = \frac{n-2}{n^2}.
  • All other squares have 0 sides painted, so the probability of picking one of those squares and then spinning a red side is 0.
  • Adding the probabilities for the separate cases gives the total probability:  \displaystyle \frac{2}{n^2}+\frac{n-2}{n^2}=\frac{1}{n}

After reading Mike’s and Alexander’s posts, I saw a much easier approach.

  • Paint all 4 edges of an nxn square, and divide each painted edge into n painted unit segments.  This creates 4 \cdot n total painted small segments.
  • Decompose the nxn original square into n^2 unit squares.  Each unit square has 4 edges giving 4 \cdot n^2 total edges.
  • Because every edge of every unit square is equally likely to be spun, the total probability of randomly selecting a smaller square and spinning a red side is \displaystyle \frac{4n}{4n^2}=\frac{1}{n}.

The dimensions of the “square” don’t seem to matter!

Oliver Wendell Holmes noted, “A mind that is stretched by a new experience can never go back to its old dimensions.”  The math after this point has the potential to stretch…

I now wondered whether this was a general property.

In the 2-dimensional square, 1-dimensional edges were painted and the square was spun to find the probability of a red edge facing.  With the originally posed cube, 2-dimensional faces were painted and the cube was tossed to find the probability of an upward-facing red face.  These cases suggest that when a cube of some dimension with edge length n is painted, is decomposed into unit cubes of the original dimension, and is spun/tossed to show a cube of one smaller dimension, then the probability of getting a painted smaller-dimensional cube of is always \displaystyle \frac{1}{n}, independent of the dimensions occupied by the cube.

Going beyond the experiences of typical middle or high school students, I calculated this probability for a 4-dimensional hypercube (a tesseract).

  • The exterior of a tesseract is 8 cubes.  Ignore the philosophical difficulty of what it means to “paint” (perhaps fill?) an entire cube.  After all, we’re already beyond the experience of our 3-dimensions.
  • Paint/fill all 8 cubes on the surface of the tesseract, and divide each painted cube into n^3 painted unit cubes.  This creates 8 \cdot n^3 total painted unit cubes.
  • Decompose the original tesseract into n^4 unit tesseracts.  Each unit tesseract has 8 cubes giving 8 \cdot n^4 total unit cubes.
  • Because every unit cube on every unit tesseract is equally likely to be “rolled”, the total probability of randomly selecting a smaller tesseract and rolling a red cube is \displaystyle \frac{8n^3}{8n^4}=\frac{1}{n}.

The probability is independent of dimension!

More formally,

The exterior of a d-dimensional hypercube with edge length n is painted red and is chopped into n^d unit d-dimensional hypercubes.  The latter are put into a bag of sufficient dimension to hold them and thoroughly mixed up.  A unit d-dimensional hypercube is withdrawn at random from the bag and tossed.  The probability that the unit d-dimensional hypercube lands with a red (d-1)-dimensional hypercube showing is \displaystyle \frac{1}{n} .

  • The exterior of a d-dimensional hypercube is comprised of 2d (d-1)-dimensional hypercubes of dimension (d-1).  Paint/fill all 2d surface hypercubes and divide each painted (d-1)-dimensional hypercube into n^{d-1} painted unit hypercubes.  This creates 2d \cdot n^{d-1} total painted unit hypercubes.
  • Decompose the original tesseract into n^d unit d-dimensional hypercubes.  Each unit d-dimensional hypercube has 2d surface (d-1)-dimensional hypercubes giving 2d \cdot n^d total surface unit d-dimensional hypercubes.
  • Because every unit (d-1)-dimensional hypercube on the surface of every unit d-dimensional hypercube is equally likely to be “rolled”, the total probability of randomly selecting a unit d-dimensional hypercube and rolling a (d-1)-dimensional red-painted hypercube is \displaystyle \frac{2d \cdot n^{d-1}}{2d \cdot n^d}=\frac{1}{n}.

I hope you can now return to something close to your old dimensions.

Inscribed Right Angle Proof Without Words

Earlier this past week, I assigned the following problem to my 8th grade Geometry class for homework.  They had not explored the relationships between circles and inscribed angles, so I added dashed auxiliary segment AD as a hint.

What follows first is the algebraic solution I expected most to find and then an elegant transformational explanation one of my students produced.

Given circle A with diameter BC and point D on the circle.  Prove triangle BCD is a right triangle.

After some initial explorations on GeoGebra sliding point D around to discover that its angle measure was always 90^{\circ} independent of the location of D, most successful solutions recognized congruent radii AB, AC, and AD, creating isosceles triangles CAD and BAD.  That gave congruent base angles x in triangle CAD, and y in BAD.

The interior angle sum of a triangle gave (x)+(x+y)+(y)=180^{\circ}, or m \angle CDB = x+y = 90^{\circ}, confirming that BCD was a right triangle.

Then, one student surprised us.  She marked the isosceles base angles as above before rotating \Delta BCD 180^{\circ} about point A.

Because the diameter rotated onto itself, the image and pre-image combined to form an quadrilateral with all angles congruent.  Because every equiangular quadrilateral is a rectangle, M had confirmed BCD was a right triangle.

I don’t recall seeing M’s proof before, but I found it a delightfully elegant application of quadrilateral properties.  In my opinion, her rotation is a beautiful proof without words solution.

Encourage freedom, flexibility of thought, and creativity, and be prepared to be surprised by your students’ discoveries!


Topic: Fiduciary

Fee-Only 101 – What does it mean?

Is your financial advisor fee-only or fee-based? Do you know the difference? If not, you may be shocked to find out how your advisor makes money off you.
Fiduciary 101: What you need to know - book with 'Fiduciary Duty' on the cover and a courtroom gavel

Fiduciary 101: What you need to know

What does 'Fiduciary' mean? Does your financial advisor have a fiduciary responsibility to you? If not, you should be concerned. Knowing this simple word and what it means could be the difference between working your entire life and an early retirement. Read our latest article to learn the facts.
broker vs ria - which advisor is a fiduciary

Broker vs RIA: Which one serves you?

Broker vs RIA - How are they compensated? The answer can be very scary. Read this article to understand how your adviser makes money off your life savings.


Neologism: Clandestine institutional knowledge

Jan 14 2020

clandestine institutional knowledge - The phenomenon in which everybody knows the documentation is wrong and people are so pissed off at said documentation that they don't ever bother to try to fix it.  Instead new hires have to play Indiana Jones to find the two people left in the organization who have any working knowledge of the thing and beg to be trained up so they can actually do their jobs.  Normally, the newly trained individual doesn't bother to update the documentation, either.

footnote: Most of the time, nobody has the access to update the documentation anymore, which is why nobody ever bothers to fix it.

Neologism: Tumbleweed mode

Jan 10 2020

tumbleweed mode - noun phrase - The phenomenon in which all official support forums for something are either abandoned (no activity for a protected period of time), or any posts that aren't lowball questions (such as "Where's the FAQ?" or replies to release announcements) are utterly ignored (meaning, actual technical support questions).

The overall state of telecommunications.

Dec 08 2019

I'm writing this article well before the year 2020.ev starts, mostly due to the fact that Twitter's search function is possibly the worst I've ever seen and this is probably my last chance to find the post in question to refer back to.

Late in November of 2019.ev a meme was going around birbsite, "Please quote this tweet with a thing that everyone in your field knows and nobody in your industry talks about because it would lead to general chaos."  Due to the fact that I was really busy at work at the time I didn't have a chance to chime in, but then an old friend of mine (and, through strange circumstances, co-worker for a time) told an absolute, unvarnished truth of the telecom industry: "Telecommunications as a whole, which also encompasses The Internet, is in a constant state of failure and just in time fixes and functionally all modern communication would collapse if about 50 people, most of which are furries, decided to turn their pager off for a day."

I don't know of any words in the English language to adequately express how true this statement is.  He's serious as the proverbial heart attack.  For a brief period of time, one solar year almost to the minute in fact, I worked for a telecommunications company in Virginia that no longer exists for reasons that are equal parts fucked up and illegal.  The company was bought out and dismantled roughly a year after I escaped by Zander's employer at the time, and seeing as how this was about fifteen years ago as you read this, I guess I can talk in public about it.

tl;dr - If you value your physical and mental health, don't work in telecom.

Have you tried turning it off and back on again?

May 25 2019

Neologism: @here grenade

Apr 25 2019

@here grenade - noun phrase - The act of tagging a message @here (meaning, everyone) in a crowded Slack channel (users >= 100), causing everyone who's busy but monitoring to drop whatever they're doing and flame you for bothering them by messaging @here.  Normally done by a user trying to get a response to a maximum severity ticket that's been ignored for longer than the SLA.

Example: "PFY threw an @here grenade into the #tech-support channel because the border router was on fire and the admins on call were ignoring their pagers.  He got kicked but at least the outage is over."

Neologism: Proper channels excise tax

Apr 16 2019

Proper channels excise tax - noun phrase - The markup paid on commonplace things when you go through proper channels at work to do something rather than going rogue, buying it yourself and filing an expense report.  For example, a flight from Chicago to Boston might cost $176us if you paid for it yourself, but by using your employer's internal processes and vendors the cost of the same flight is closer to $630us.

Neologism: Taxonomic debt

Apr 06 2019

Taxonomic debt - noun phrase - The time you spend learning arbitrary jargon at a new job.

Source: Bradford Stephens

Neologism: Sharkfinning

Nov 03 2018

sharkfinning - verb, gerund - Learning something from scratch in an entirely hands-on way, which is to say, "Swimming with the sharks."  When you don't know what you're doing or how to do it, but you have a job to do.

Life and times.

Oct 14 2018

Long time readers are probably wondering where I've been lately.  The answer is kind of long and is worth a post all on its own.  The short version of the story is, work's been eating me alive lately.  This is our busiest time of year and it's been all hands on deck for a couple of weeks now.  In point of fact, last week was our quarterly all-hands meeting, where everybody on my team was flown into town for a solid week of meetings.  All day, every day.  Most of my visible activity lately took the form of parts of my exocortex running on automatic with some hit-and-run posting while waiting for the coffee maker at work to top me up in between meetings.

This also means that I haven't had a whole lot of patience for interacting with people.  Not in the sense that people can feel frustrated with other people or their actions, but in the sense that interacting with people in a meaningful way - having a real conversation - takes more compute cycles than I have available right now.  After fourteen hours in a conference room with 40 other people, not only am I out of social, but I'm mentally exhausted.


What’s on your Winter #CTBucketList?

WASHINGTON, DC — Lawyers for two Iraqis with ties to the US military who had been granted visas to enter the United States have filed a lawsuit against President Donald Trump and the US government after they were detained when they arrived in New York Friday.

The lawsuit could represent the first legal challenge to Trump’s controversial executive order, which indefinitely suspends admissions for Syrian refugees and limits the flow of other refugees into the United States by instituting what the President has called “extreme vetting” of immigrants.

Trump’s order also bars Iraqi citizens, as well as people from six other Muslim-majority nations, from entering the US for 90 days, and suspends the US Refugee Admissions Program for 120 days until it is reinstated “only for nationals of countries for whom” members of Trump’s Cabinet deem can be properly vetted.

According to court papers, both men legally were allowed to come into the US but were detained in accordance with Trump’s move to ban travel from several Muslim-majority nations.

One of the men, Hameed Khalid Darweesh, who worked as an interpreter for the US during the Iraq War, was released from detention early Saturday afternoon.

“America is the land of freedom,” Darweesh told reporters at the airport shortly after his release. “America is the greatest nation.”

Asked about Trump, Darweesh said, “I like him, but I don’t know.” He added that he was separated from his family Friday at the airport.

A source with knowledge of the case confirmed Darweesh will be allowed into the US due to provisions in Trump’s order that allow the State and Homeland Security departments to admit individuals into the US on a case-by-case base for certain reasons, including when the person is already in transit and it would cause undue hardship and would not pose a threat to the security of the US.

The suit said Darweesh held a special immigrant visa, which he was granted the day of Trump’s inauguration on January 20, due to his work for the US government from 2003 to 2013.

The other Iraqi man named as a plaintiff in the suit is Haider Sameer Abdulkaleq Alshawi, who was still being held at JFK as of early Saturday afternoon. After meeting Darweesh for his release, Rep. Jerrold Nadler, D-New York, said he would work with Rep. Nydia Velazquez, D-New York, to secure the release of Alshawi and 10 other people still detained.

Nadler and Velazquez railed against Trump’s order and pledged continued action.

“This should not happen in America,” Velazquez said following Darweesh’s release. “One by one, street by street, if we have to go to court, we will fight this anyplace, anywhere.”

The status of the 11 people Nadler mentioned was not immediately clear.

‘The executive order is unlawful’

The lawsuit said the US granted Alshawi a visa earlier this month to meet with his wife and son, whom the US already granted refugee status for her association with the US military.

“Because the executive order is unlawful as applied to petitioners, their continued detention based solely on the executive order violates their Fifth Amendment procedural and substantive due process rights,” the lawyers argue in court papers.

Court papers said Customs and Border Protection authorities did not allow the lawyers to meet with the men and told them to try reaching Trump. Nadler and Velázquez said they attempted to speak to Darweesh and Alshawi at JFK’s Terminal 4 earlier Saturday but were denied.

“When Mr. Darweesh’s attorneys approached CBP requesting to speak with Mr. Darweesh, CBP indicated that they were not the ones to talk to about seeing their client. When the attorneys asked, ‘Who is the person to talk to?’ the CBP agents responded, ‘Mr. President. Call Mr. Trump,'” the court papers read.

Lawyers for Iraqi refugees filed the suit in federal court arguing Darweesh and Alshawi were being unlawfully held “solely pursuant” to an executive order issued on January 27, 2017.

Arguing that their clients have “valid entry documents,” they say the were blocked from exiting JFK and detained.

One of the attorneys, Mark Doss of the International Refugee Assistance Project, told CNN his clients knew they had to get to the US as soon as possible so they boarded the first flight they could.

The two men had been allowed to make phone calls. They do not know each other, and it is unclear if they were being held together or separately, or if they were being kept in a holding cell, according to Doss. One man traveled from Sweden and one from Iraq.

Immigration-rights groups, including the National Immigration Law Center and the ACLU, are representing the Iraqi men. Lawyers for the Iraqis are aiming to file a class-action lawsuit on behalf of other refugees.

“Our courageous plaintiff and countless others risked their lives helping US service members in Iraq. Trump’s order puts those who have helped us in harm’s way by denying them the safe harbor they have been promised in the United States,” said Karen Tumlin, the legal director of the NILC.

The lawsuit was earlier reported by The New York Times.

Legality questioned

Trump’s executive order, titled “Protection Of The Nation From Foreign Terrorist Entry Into The United States,” makes good on his longtime campaign promise to tighten borders and halt certain refugees from entering the United States.

“I am establishing new vetting measures to keep radical Islamic terrorists out of the United States of America,” Trump said during the signing at the Pentagon. “We don’t want them here.”

But the order immediately raised questions about its legal standing.

“What this order does is take people who played by the rules who have been rigorously vetted and slams the door of the country in their face because of their religion,” said David Leopold, an immigration attorney and the former president of the American Immigration Lawyers Association.

But Stephen Yale-Loehr, a professor of immigration law practice at Cornell Law School, said Trump’s order may have merit.

“The plaintiffs are alleging a variety of reasons why they should not be detained under the executive order. Their arguments include the equal protection clause of the Constitution and a provision in the immigration law that prohibits national origin discrimination,” Yale-Loehr said. “However, presidents have wide discretion on immigration, because immigration touches national sovereignty and foreign relations. Courts tend to defer to whatever a president declares on immigration. I think the administration could win.”

Progressives outraged

Democrats and liberal-leaning groups continued to denounce Trump’s order Saturday as racist.

“The first two refugees kept out of America by President Trump’s order are Iraqis who helped US troops survive during the war,” Rep. Jim Cooper, D-Tennessee, said on Twitter. “These men risked their lives for years to keep our troops safe. One was with the 101st Airborne. This is no way to treat our allies.”

Vermont Sen. Bernie Sanders called the action an “anti-Muslim order” that “plays into the hands of fanatics wishing to harm America.”

And the ACLU repeatedly referred to the executive order as Trump’s “Muslim ban.”

Travelers turned away

Sarah Assali from Allentown, Pennsylvania, told CNN Saturday that the US government stopped six of her Syrian family members at Philadelphia International Airport and sent them back to Doha, Qatar, the origin city of their flight.

Assali said her family members were persecuted Christians and did not speak very good English, so they did not fully understand why they were being put on a plane back to the Middle East.

Additionally, some US-bound travelers who were either refugees or from countries specified in Trump’s executive ban have been turned back from Cairo International Airport, according to a Cairo airport official who is not authorized to speak to the media.

The official said they were required to follow the instructions of the United State in the matter. He said “(The airport personnel) have no choice but to follow orders. I don’t want to blame anyone here.”

“This is a new era we are witnessing,” the official said.

When pressed about the exact protocol on who to turn back, including whether or not US green card holders from the seven Muslim-majority countries were banned entry under the order, the official said, “I don’t know, we are waiting for more clarifications.” He also would not specify how many were being turned back.

Iran responded to Trump’s action by banning all US citizens from entering the country.


Was Freud Really An Atheist?

FreudFreud’s daughter Anna, the only child to carry on his work, once said . . . “If you want to know my father, don’t read his biographers, read his letters.” A careful reading of his letters reveals some rather surprising—if not perplexing—material. First, Freud frequently quoted from the Bible, both the Old and New Testaments. In his autobiography Freud writes: “My early familiarity with the Bible story . . . had, as I recognized much later, an enduring effect upon the direction of my interest.” Second, letters written through his life are replete with words and phrases such as “I passed my examinations with God’s help”; “if God so wills”; “the good Lord”; “taking the Lord to task”; “into the keeping of the Lord”; “until after the Resurrection”; “science seems to demand the existence of God”; “God’s judgment”; “God’s will”; “God’s grace”; “God above”; “if someday we meet above”; “in the next world”; “my secret prayer.”

In a letter to Oskar Pfister [a Swiss pastor and psychoanalyst], Freud writes that Pfister was “a true servant of God” and was “in the fortunate position to lead (others) to God.” What does this mean? Can we not dismiss all this a merely figures of speech—common in English as well as in German? Yes, if it were anyone but Freud. But Freud insisted even a slip of the tongue had meaning.

The preoccupation continues until his last book, Moses and Monotheism, written . . . when he was in his eighties. Why? Why couldn’t he put the question to rest? If he had all the answers, why did the question of God’s existence continue to preoccupy him? Perhaps C. S. Lewis would say we can never explain away God. Nor can we find rest until that deep-seated desire (experienced by both Freud and Lewis) is satisfied.

–Dr. Armand M. Nicholli, Jr.,
The Question of God

Looking for Signs of God

A great many arguments about God—God’s existence, God’s nature, God’s actions in the world—run the risk of being like pointing a flashlight toward the sky to see if the sun is shining. Is all too easy to make the mistake of speaking and thinking as though God (if there is a God) might be a being, an entity, within our world, accessible to our interested study in the same sort of way we might studying music or mathematics, open to our investigation by the same sort of techniques will use for objects and entities within our world.

imagesWhen Yuri Gagari, the first Soviet cosmonaut, landed after orbiting the Earth a few times, he declared that he had disproved the existence of God. He had been up there, he said, and had seen no sign of him. Some Christians pointed out that Gagarin had seen plenty of signs of God, if only the cosmonaut known how to interpret them. The difficulty is that speaking of God in anything like the Christian sense is like staring into the sun it’s dazzling. It’s easier, actually, to look away from the sun itself and to enjoy the fact that once it’s well and truly risen, you can see everything else clearly.

–N. T. Wright,
Simply Christian


Modularity Matters: Learning Invariant Relational Reasoning Tasks

Former CNN models fully activate(filly distributed features) for a single input showing poor performance on invariant relational reasoning.

The reason former CNN models are poor at invariant reasoning is interference problem that learning each pattern interfere with each other while a model try to learn many patterns. To minimize this problem, this paper suggest mixture of experts architecture(MoE). MoE architecture consists of two parts: Individual expert networks(E_1, ... E_n) and Gater network(G) that weight the output of each expert network.

y = \sum_{i=1}^E G[i]E_i

In this paper, each expert network consists of multiple residual blocks and the architecture is called Residual Mixture Network(ResMixNet).

This paper suggested two tasks to test invariant relational reasoning. First is MNIST Parity dataset which is to predict the parity(both even or both odd) of a pair of numbers in a image. Second is colorized Pentomino dataset which is to learn whether all the Pentomino sprites in an image belong to the same class. ResMixNet achieved the best result on both of these tasks compared to traditional CNN models.

Modularity Seems quite naive. There should be further development.

Jo, Jason, Vikas Verma, and Yoshua Bengio. “Modularity Matters: Learning Invariant Relational Reasoning Tasks.” arXiv preprint arXiv:1806.06765 (2018).

© 2017. by isme2n

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