A kaleidoscope works by mirrors.
Specifically an equilateral triangle of mirrors which present reflections and reflections of reflections. Here is the inner core. You can drag the red point to rotate the kaleidoscope. You can also drag the vertices of the original triangles.
The problem – find the triangle with minimal perimeter that can be inscribed in a given triangle.
The solution: reflection helps reduce the problem to a single parameter.
The answer: is geometrically appealing.
Can you solve it without using the hint?
Given a circle with center A and a point C, here are three related ways of constructing Pascal’s Limacon:
It is the isotomic (look it up) of the circle with pole C.
It is the conchoid (look it up) of a circle centered at A whose circumference passes through C.
It is an epitrochoid (look it up) formed by a circle of equal radius rolling around the original.
This app constitutes a visual proof of the above, and depends on the fact that the composition of reflections in two parallel lines is equivalent to a translation of twice the distance between the lines.
A simple box solar cooker works by reflecting sunlight from its lid into the box.
Can you work out a relationship between the angle of the sunlight and the best angle to open the lid?
Can you prove it?
The solar concentration ratio of a solar cooker is the ratio of the amount of sunlight concentrated on the target to the size of the target.
This model presents the best situation for a given lid angle, and lets you see how much light can be captured at that lid angle.
The question is: what lid angle captures the most sunlight?
One definition of an ellipse is as the locus of points the sum of whose distance from two fixed points (the foci) is constant.
This gives you a way to draw the ellipse with two pins and a piece of string.
Here we relate that definition to the focal property: that light emanating from one focus reflects to the other focus.
Light from C reflects in the tangent to the circle at point B.
Observe the trace as B rotates around the circle. While the individual rays move, the phantom curve - the caustic - does not.
The caustic is the mathematical envelope of the reflected rays, which is to say that each reflected ray is tangential to the envelope curve.
At any moment, each ray is moving with a combination of rotation and translation. Each point on the ray is undergoing a translation. Points on the ray which lie on the caustic are moving along teh length of the ray, and hence their motion is not visible. Which is why the caustic looks stationary while everything else moves.
Drag B to find the maximum angle of reflection.
The circumcircle of ABC is the locus of all points which subtend the same angle on chord AC.
In general G is inside the circumcircle and so angle AGC < angle ABC.
The exception is when B=G.
The hour hand of this clock defines the radiant point.
The minute hand gives the target for a single ray.
The second hand gives the target for a family of rays.
You can reflect the radiant point in the second hand to observe its relationship to the cusps of the caustic.