  We study the geometry of algebraic numbers in the complex plane, and their
Diophantine approximation, aided by extensive computer visualization. Motivated
by these images, called algebraic starscapes, we describe the geometry of the
map from the coefficient space of polynomials to the root space, focussing on
the quadratic and cubic cases. The geometry describes and explains notable
features of the illustrations, and motivates a geometric-minded recasting of
fundamental results in the Diophantine approximation of the complex plane. The
images provide a case-study in the symbiosis of illustration and research, and
an entry-point to geometry and number theory for a wider audience. The paper is
written to provide an accessible introduction to the study of homogeneous
geometry and Diophantine approximation.
  We investigate the homogeneous geometry of root and coefficient spaces under
the natural $\operatorname{PSL}(2;\mathbb{C})$ action, especially in degrees 2
and 3. We rediscover the quadratic and cubic root formulas as isometries, and
determine when the map sending certain families of polynomials to their complex
roots (our starscape images) are embeddings.
  We consider complex Diophantine approximation by quadratic irrationals, in
terms of hyperbolic distance and the discriminant as a measure of arithmetic
height. We recover the quadratic case of results of Bugeaud and Evertse, and
give some geometric explanation for the dichotomy they discovered (Bugeaud, Y.
and Evertse, J.-H., Approximation of complex algebraic numbers by algebraic
numbers of bounded degree, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 8 (2009),
no. 2, 333-368). Our statements go a little further in distinguishing
approximability in terms of whether the target or approximations lie on
rational geodesics.
  The paper comes with accompanying software, and finishes with a wide variety
of open problems.
