User:Harry Princeton/Circle Packings and Ambo Tilings

A summary of my investigations in circle packing and ambo tilings.

Packings in the plane

Identical circles in a hexagonal packing arrangement, the densest packing possible.

Triangular packing through natural arrangement of equal circles with transitions to an irregular arrangement of unequal circles.

In two dimensional Euclidean space, Joseph Louis Lagrange proved in 1773 that the highest-density lattice arrangement of circles is the hexagonal packing arrangement, in which the centres of the circles are arranged in a hexagonal lattice (staggered rows, like a honeycomb), and each circle is surrounded by 6 other circles. The density of this arrangement is

$\eta _{h}={\frac {\pi {\sqrt {3}}}{6}}\approx 0.9069.$

Hexagonal packing of equal circles was found to fill a fraction Pi/Sqrt[12] ≃ 0.91 of area—which was proved maximal for periodic packings by Carl Friedrich Gauss in 1831. Later, Axel Thue provided the first proof that this was optimal in 1890, showing that the hexagonal lattice is the densest of all possible circle packings, both regular and irregular. However, his proof was considered by some to be incomplete. The first rigorous proof is attributed to László Fejes Tóth in 1940.

At the other extreme, Böröczky demonstrated that arbitrarily low density arrangements of rigidly packed circles exist.

Uniform packings

There are 11 circle packings based on the 11 uniform tilings of the plane. In these packings, every circle can be mapped to every other circle by reflections and rotations. The hexagonal gaps can be filled by one circle and the dodecagonal gaps can be filled with 7 circles, creating 3-uniform packings. The truncated trihexagonal tiling with both types of gaps can be filled as a 4-uniform packing. These operations are conjugate to polygon dissections and act as dual insets. The snub hexagonal tiling has two mirror-image forms.

The ambo version of the circle packing of the 12 fundamental uniform tilings. Three are regular tilings, eight are uniform tilings, and one is 2-uniform. It consists of smaller triangles, squares, hexagons, and dodecagons whose vertices are at original midpoints. Mini-vertex polygons correspond to each vertex in the original tiling. These are homeomorphic to the left circle packings.

A diagram showing a circle of radius 1/2 inscribed in all planigons, semiplangions, and unusable triangles. Inside these circles, the corresponding vertex-type mini polygons are lightly shaded (inscribed). They are half the size of vertex-type polygons. Everything is to scale (2560×1280).

In fact, the fundamental domains of such Archimedean circle packings are the planigons. Moreover, a circle of radius $1/2$ can be inscribed in every planigon for $k$-uniform circle packings, because the dual vertex figure consists of segments of length $1/2$ emanating from the vertex. This follows from the fact that the edges of the dual uniform tilings intersect the edges of the uniform tilings orthogonally at midpoints (hence the $1/2$), as the Conway operation of dualization (or ortho) involves connecting the centroids to the midpoints of convex regular polygons. This also works for semiplanigons as well, as seen below (all to scale - caption discrepancies generated by Asymptote):

Another way for seeing this for all semiplangions except the tie kite (V3.4.3.12) is as follows: a skew quadrilateral (V3².4.12) is the result of removing a sixth of a planigonal regular hexagon ('tiny triangle') from the $60^{\circ }$ angle of the planigonal scalene right triangle (kisrhombille tiling); an isosceles trapezoid (V3².6²), planigonal Floret pentagon, planigonal rhombus, and planigonal equilateral triangle all result from adding a number tiny triangles to a planigonal regular hexagon (Floret pentagonal, rhombille, deltille tilings); and a right trapezoid (V3.4².6) is the result of adding a tiny triangle to a prismatic pentagon (prismatic pentagonal tiling). These external tiny triangular additions (or removal) preserve the cocyclic nature of the polygons, and the inscribed circle's radius remains the same in all cases:

k-Uniform Circle Packing Examples

The 5th Krotenheerdt 4-uniform tiling [3².4.12; 3².12; 3².4.3.4; 3⁶] (a gem-like tiling whose dual is gem-like as well) is circle-packed below, with each circle and corresponding dual planigon/semiplangion sharing the same color. Also shown are the original tiling, the ambo tiling, and the dual tiling. The uniform ambo tiling is made by connecting the midpoints of all regular polygons, and the spaces in between are the vertex-figure polygons (1-D duals to the planigons/semiplangions by interchanging vertices and edges), and it is homeomorphic to the uniform circle packing by means of circumscribing vertex figure polygons (and plasmolysing regular polygons), or conversely by taking the convex hulls of the gaps between the circles (and inscribing vertex-figure polygons with its own vertices at tangential points). The vertex-figure polygons are also colored according to the corresponding plangions/semiplanigons. All images and colors coincide.

A Circle Packing, Dual, and Ambo of the 4-uniform [3².4.12; 3².12; 3².4.3.4; 3⁶] tiling.

4-Uniform Tiling	4-Uniform Circle Packing	4-Uniform Dual Tiling	4-Uniform Ambo Tiling	4-Uniform Tiling
[32.4.12; 32.12; 32.4.3.4; 36]	C[32.4.12; 32.12; 32.4.3.4; 36]	d=V[32.4.12; 32.12; 32.4.3.4; 36]	a[32.4.12; 32.12; 32.4.3.4; 36]	[32.4.12; 32.12; 32.4.3.4; 36]

A circle packing and ambo operation is done on the 7-uniform Krotenheerdt 2 tiling, All circles, colors, planigons/semiplangions, polygons, and vertex figure polygons coincide, with the same scale.

A Circle Packing, Dual, and Ambo of the 7-uniform [3⁶; 3³.4²; 3².4.3.4; 3⁴.6; 3.4².6; 3².4.12; 4.6.12] tiling.

7-Uniform Tiling	7-Uniform Circle Packing	7-Uniform Dual Tiling	7-Uniform Ambo Tiling	7-Uniform Tiling
[36; 33.42; 32.4.3.4; 34.6; 3.42.6; 32.4.12; 4.6.12]	C[36; 33.42; 32.4.3.4; 34.6; 3.42.6; 32.4.12; 4.6.12]	d=V[36; 33.42; 32.4.3.4; 34.6; 3.42.6; 32.4.12; 4.6.12]	V[36; 33.42; 32.4.3.4; 34.6; 3.42.6; 32.4.12; 4.6.12]	[36; 33.42; 32.4.3.4; 34.6; 3.42.6; 32.4.12; 4.6.12]

Finally, the same treatment is done to a 92-uniform tiling, which consists of 14 vertex figures (and whose dual consists of 14 plangions). This is the maximum number of types of vertices (resp. plangions) allowed in any uniform tiling (resp. dual uniform tiling). All circles, colors, planigons/semiplangions, polygons, and vertex figure polygons coincide, but the scale is shrunk to $1/2$.

A Circle Packing, Dual, and Ambo of a 92-uniform tiling containing 14 vertex-figure polygons.

92-Uniform Tiling	92-Uniform Circle Packing	92-Uniform Dual Tiling	92-Uniform Ambo Tiling	92-Uniform Tiling
[92-uniform tiling]	C[92-uniform tiling]	d=V[92-uniform tiling]	a[92-uniform tiling]	[92-uniform tiling]

Finally, we could have many non-uniform radial circle packings, but with only two types of circles (V3³.4²; V3².4.3.4). One of them is shown below, along with the homeomorphic ambo tiling:

Circle Packing

This 2-uniform tiling can be used as a circle packing. Cyan circles are in contact with 3 other circles (1 cyan, 2 pink), corresponding to the V3.12² planigon, and pink circles are in contact with 4 other circles (2 cyan, 2 pink), corresponding to the V3.4.3.12 planigon. It is homeomorphic to the ambo operation on the tiling, with the cyan and pink gap polygons corresponding to the cyan and pink circles (one dimensional duals to the respective planigons). Both images coincide.

Circle Packing	Ambo

Circle Packing

This 2-uniform tiling can be used as a circle packing. Cyan circles are in contact with 3 other circles (2 cyan, 1 pink), corresponding to the V4.6.12 planigon, and pink circles are in contact with 4 other circles (1 cyan, 2 pink), corresponding to the V3.4.6.4 planigon. It is homeomorphic to the ambo operation on the tiling, with the cyan and pink gap polygons corresponding to the cyan and pink circles (mini-vertex configuration polygons; one dimensional duals to the respective planigons). Both images coincide.

C[3.4.6.12]	a[3.4.6.12]

Circle Packings

These 2-uniform tilings can be used as a circle packings.

In the first 2-uniform tiling (whose dual resembles a key-lock pattern): cyan circles are in contact with 5 other circles (3 cyan, 2 pink), corresponding to the V3³.4² planigon, and pink circles are also in contact with 5 other circles (4 cyan, 1 pink), corresponding to the V3².4.3.4 planigon. It is homeomorphic to the ambo operation on the tiling, with the cyan and pink gap polygons corresponding to the cyan and pink circles (mini-vertex configuration polygons; one dimensional duals to the respective planigons). Both images coincide.

In the second 2-uniform tiling (whose dual resembles jagged streams of water): cyan circles are in contact with 5 other circles (2 cyan, 3 pink), corresponding to the V3³.4² planigon, and pink circles are also in contact with 5 other circles (3 cyan, 2 pink), corresponding to the V3².4.3.4 planigon. It is homeomorphic to the ambo operation on the tiling, with the cyan and pink gap polygons corresponding to the cyan and pink circles (mini-vertex configuration polygons; one dimensional duals to the respective planigons). Both images coincide.

Circle Packings of and Ambo Operations on Two Pentagonal Isoperimetric 2-dual-uniform tilings.

C[33.42; 32.4.3.4]1	a33.42; 32.4.3.4]1	C[33.42; 32.4.3.4]2	a[33.42; 32.4.3.4]2