##### Document Text Contents

Page 1

A Mathematical Coloring Book

by Marshall Hampton

Dedicated to Violet Hampton

Version 0.94

Copyright 2009

Send comments and suggestions to [email protected]

Page 2

Figure 1: Schlegel projection of an icosahedron

Page 19

Figure 18: Archimedean and Exponential Spirals

Page 20

Figure 19: Cobweb diagram of the logistic map # 1.

Page 37

be 3264 real conics to five circles. I still don’t know the answer, but this picture of

some of the real tangent conics to five circles on a regular pentagon might help give

some intuition...

13. Geodesics in the unit disk model of hyperbolic space. This is an approximation to a

tiling by pentagons in which four pentagons meet at each vertex.

14. Part of the hyperbolic plane tiled with stars. In hyperbolic space, each of these stars

is the same size and the edges are ”straight” - i.e. geodesics.

15. The Gröbner fan of the ideal < x5 − y4, y5 − z4, z5 − x4 >.

16. Gröbner fan of the ideal < x3 − y2, y3 − z2, z3 − x2 >

17. Gröbner fan of the vortex problem ideal defined by

−6xyz + xy + xz + 6yz − y2 − z2,

−6xyz + xy + 6xz + yz − x2 − z2,

−6xyz + 6xy + xz + yz − x2 − y2

The variables are the squares of the distances between the three vortices. This equa-

tions determine the stationary configurations of three equal-strength vortices.

18. An Archimedean spiral (r = t) and an exponential spiral (r = e15.73t; chosen arbitrarily

to make things look nicer).

19. Cobweb diagram for the logistic map f(x) = 3.9x(1− x). The graphs of the function

f(x) and its first two iterates f(f(x)) and f(f(f(x))) are plotted.

20. Cobweb diagram for the logistic map f(x) = 3.9605x(1−x). The graphs of the function

f(x), f(f(x)) and f(f(f(f(x)))) are plotted. This parameter for the logistic map is

sitting in a small period-4 window.

21. Equipotential lines for the equal-mass three-body problem. The levels are not equally

spaced in value.

22. Orbits in the planar circular restricted three-body problem with µ = .5 (equal mass

primaries).

23. Sierpinski triangle. One of the first fractal structures considered, it and similar fractals

are important in the topological classification of continua. It has Hausdorff dimension

log(3)/ log(2) ≈ 1.585.

24. Nested circles and Koch snowflakes (finite iterates). The Koch snowflake is one of the

first fractals ever constructed (1906).

Page 38

25. Inverse images of two circles, radii 1 and 1/4, under a quadratic map. Image is rotated

90 degrees for a better aspect ratio on the page (so the imaginary axis is horizontal).

The map is z → z2 − .99 + 0.1I.

26. Inverse images of three circles, radii 1, 1/2, and 1/4, under a quadratic map. Image

is rotated 90 degrees for a better aspect ratio on the page (so the imaginary axis is

horizontal). The map is z → z2 − 0.8 + .156I.

27. Hypotrochoid - ? (x(t), y(t)) = (8cos(t) + 8cos(17t/2), 8sin(t)− 8sin(17t/2)).

28. Rotationally symmetric arrangement of parallelogram tiles. The most acute angles in

the three types of tiles in this and the next three figures have angles π/7, 3π/14, and

2π/7.

29. Another rotationally symmetric arrangement of parallelogram tiles.

30. Another rotationally symmetric arrangement of parallelogram tiles.

31. Symmetric Venn diagram for 5 sets represented by ellipses. Discovered by Branko

Grunbaum in 1975.

32. “Adelaide”. A beautiful 7-set symmetric Venn diagram discovered independently by

Branko Grunbaum and Anthony Edwards.

33. Limacons r = 1 + q cos(t) with q ∈ [0, 5].

34. Lissajous curve x = cos(12t), y = sin(13t).

A Mathematical Coloring Book

by Marshall Hampton

Dedicated to Violet Hampton

Version 0.94

Copyright 2009

Send comments and suggestions to [email protected]

Page 2

Figure 1: Schlegel projection of an icosahedron

Page 19

Figure 18: Archimedean and Exponential Spirals

Page 20

Figure 19: Cobweb diagram of the logistic map # 1.

Page 37

be 3264 real conics to five circles. I still don’t know the answer, but this picture of

some of the real tangent conics to five circles on a regular pentagon might help give

some intuition...

13. Geodesics in the unit disk model of hyperbolic space. This is an approximation to a

tiling by pentagons in which four pentagons meet at each vertex.

14. Part of the hyperbolic plane tiled with stars. In hyperbolic space, each of these stars

is the same size and the edges are ”straight” - i.e. geodesics.

15. The Gröbner fan of the ideal < x5 − y4, y5 − z4, z5 − x4 >.

16. Gröbner fan of the ideal < x3 − y2, y3 − z2, z3 − x2 >

17. Gröbner fan of the vortex problem ideal defined by

−6xyz + xy + xz + 6yz − y2 − z2,

−6xyz + xy + 6xz + yz − x2 − z2,

−6xyz + 6xy + xz + yz − x2 − y2

The variables are the squares of the distances between the three vortices. This equa-

tions determine the stationary configurations of three equal-strength vortices.

18. An Archimedean spiral (r = t) and an exponential spiral (r = e15.73t; chosen arbitrarily

to make things look nicer).

19. Cobweb diagram for the logistic map f(x) = 3.9x(1− x). The graphs of the function

f(x) and its first two iterates f(f(x)) and f(f(f(x))) are plotted.

20. Cobweb diagram for the logistic map f(x) = 3.9605x(1−x). The graphs of the function

f(x), f(f(x)) and f(f(f(f(x)))) are plotted. This parameter for the logistic map is

sitting in a small period-4 window.

21. Equipotential lines for the equal-mass three-body problem. The levels are not equally

spaced in value.

22. Orbits in the planar circular restricted three-body problem with µ = .5 (equal mass

primaries).

23. Sierpinski triangle. One of the first fractal structures considered, it and similar fractals

are important in the topological classification of continua. It has Hausdorff dimension

log(3)/ log(2) ≈ 1.585.

24. Nested circles and Koch snowflakes (finite iterates). The Koch snowflake is one of the

first fractals ever constructed (1906).

Page 38

25. Inverse images of two circles, radii 1 and 1/4, under a quadratic map. Image is rotated

90 degrees for a better aspect ratio on the page (so the imaginary axis is horizontal).

The map is z → z2 − .99 + 0.1I.

26. Inverse images of three circles, radii 1, 1/2, and 1/4, under a quadratic map. Image

is rotated 90 degrees for a better aspect ratio on the page (so the imaginary axis is

horizontal). The map is z → z2 − 0.8 + .156I.

27. Hypotrochoid - ? (x(t), y(t)) = (8cos(t) + 8cos(17t/2), 8sin(t)− 8sin(17t/2)).

28. Rotationally symmetric arrangement of parallelogram tiles. The most acute angles in

the three types of tiles in this and the next three figures have angles π/7, 3π/14, and

2π/7.

29. Another rotationally symmetric arrangement of parallelogram tiles.

30. Another rotationally symmetric arrangement of parallelogram tiles.

31. Symmetric Venn diagram for 5 sets represented by ellipses. Discovered by Branko

Grunbaum in 1975.

32. “Adelaide”. A beautiful 7-set symmetric Venn diagram discovered independently by

Branko Grunbaum and Anthony Edwards.

33. Limacons r = 1 + q cos(t) with q ∈ [0, 5].

34. Lissajous curve x = cos(12t), y = sin(13t).