From bede4516d9c1f1e333d845e0073c851add772a90 Mon Sep 17 00:00:00 2001 From: queue-miscreant Date: Tue, 25 Mar 2025 20:19:12 -0500 Subject: [PATCH] ignore identity in double-goldberg recipes --- _freeze/posts/pentagons/3/index/execute-results/html.json | 4 ++-- posts/pentagons/3/index.qmd | 5 +++-- 2 files changed, 5 insertions(+), 4 deletions(-) diff --git a/_freeze/posts/pentagons/3/index/execute-results/html.json b/_freeze/posts/pentagons/3/index/execute-results/html.json index dd1c763..8a35217 100644 --- a/_freeze/posts/pentagons/3/index/execute-results/html.json +++ b/_freeze/posts/pentagons/3/index/execute-results/html.json @@ -1,8 +1,8 @@ { - "hash": "764b1870a88e8041a642faff49738f88", + "hash": "399c7fef9a9b6a23dbb456d4a3f37b04", "result": { "engine": "jupyter", - "markdown": "---\ntitle: \"12 Pentagons, Part 3\"\ndescription: |\n Exploring even more symmetries based on the 12 pentagons condition.\nformat:\n html:\n html-math-method: katex\njupyter: python3\ndate: \"2021-06-09\"\ndate-modified: \"2025-03-16\"\ncategories:\n - geometry\n - combinatorics\n - symmetry\n---\n\n\n\n\n\nThis is the third part in an investigation into answering the following question:\n\n> A soccer ball is a (roughly spherical) figure made of pentagons and hexagons, each meeting 3 at a point.\n [T]here are 12 pentagons...how many hexagons can there be?\n\nThe [first post](../1) in the series proved the 12 pentagons portion and investigated\n (dodecahedral) Goldberg polyhedra.\nThe [second post](../2) investigated tetrahedral Goldberg polyhedra as well as other tetrahedral solutions.\nYet more unconventional solutions are presented below.\n\n\nCombinatorics of Goldberg-Coxeter Operators\n-------------------------------------------\n\nIn the first post, we established that a solution figure *S* has feature counts\n which are parametrized entirely by $F_6$:\n\n$$\nS = \\begin{pmatrix} v \\\\ e \\\\ f \\end{pmatrix}\n = \\begin{pmatrix}\n 2F_6 + 20 \\\\\n 3F_6 + 30 \\\\\n F_6 + 12\n \\end{pmatrix}\n = \\begin{pmatrix} 2 \\\\ 3 \\\\ 1 \\end{pmatrix} F_6\n + \\begin{pmatrix} 20 \\\\ 30 \\\\ 12 \\end{pmatrix}\n$$\n\nIn the last post, it was discovered that certain solutions cannot be created easily from seed polyhedra\n in a polyhedron viewer.\nThis prompts the question of how to calculate the vertex, edge, and face counts\n without acquiring them from the output of such a program.\nFortunatelly, each of the simple GC operators (*dk*, *c*, and *w*) has a combinatoric matrix form[^1].\n\n$$\n\\underset{\n \\text{Class I}\n}{\n c = \\begin{pmatrix}\n 1 & 2 & 0 \\\\\n 0 & 4 & 0 \\\\\n 0 & 1 & 1\n \\end{pmatrix}\n} \\quad\n\\underset{\n \\text{Class II}\n}{\n dk = \\begin{pmatrix}\n 0 & 2 & 0 \\\\\n 0 & 3 & 0 \\\\\n 1 & 0 & 1\n \\end{pmatrix}\n} \\quad\n\\underset{\n \\text{Class III}\n}{\n w = \\begin{pmatrix}\n 1 & 4 & 0 \\\\\n 0 & 7 & 0 \\\\\n 0 & 2 & 1\n \\end{pmatrix}\n}\n$$\n\n[^1]:\n These matrices can be derived by closely observing the vertex, edge, and face counts of a figure before\n and after applying the operator.\n Gather the feature counts of the tetrahedron, dodecahedron, and icosahedron into a matrix.\n\n $$\n \\begin{gather*}\n f \\begin{pmatrix} | & | & | \\\\ T & D & I \\\\ | & | & | \\end{pmatrix}\n = f \\begin{pmatrix} 4 & 20 & 12 \\\\ 6 & 30 & 30 \\\\ 4 & 12 & 20 \\end{pmatrix}\n = \\begin{pmatrix} | & | & | \\\\ fT & fD & fI \\\\ | & | & | \\end{pmatrix} \\\\\n f = \\begin{pmatrix} | & | & | \\\\ fT & fD & fI \\\\ | & | & | \\end{pmatrix}\n \\begin{pmatrix} 4 & 20 & 12 \\\\ 6 & 30 & 30 \\\\ 4 & 12 & 20 \\end{pmatrix}^{-1}\n \\end{gather*}\n $$\n\n This matrix is invertible, and the feature counts for *fT*, *fD*, and *fI* can be acquired from a viewer\n implementing the operator *f*.\n\nThese operators are labelled with the classes of Goldberg polyhedra they construct\n when applied to the dodecahedron.\n\n- *dkD* = *tI* = GC(1, 1)\n- *cD* = GC(2, 0)\n- *wD* = GC(2, 1)\n\nIt's worth noting again that *dk* alternates between producing Class I and Class II solutions.\n\n\n### Diagonalization\n\n*tk* is the square of the *dk* operator.\nUnlike it, it preserves solution class.\nPowers of a matrix are more readily expressed when the matrix is diagonalized.\nDiagonalizing each of these operators shows that they have something in common.\n\n::: {#fbd64b61 .cell execution_count=2}\n\n::: {.cell-output .cell-output-display .cell-output-markdown execution_count=46}\n$$\n \\begin{align*}\n dk &= \\begin{pmatrix}0 & 2 & 0\\\\0 & 3 & 0\\\\1 & 0 & 1\\end{pmatrix} = \\textcolor{red}{\\begin{pmatrix}-1 & 0 & 2\\\\0 & 0 & 3\\\\1 & 1 & 1\\end{pmatrix}}\\begin{pmatrix}0 & 0 & 0\\\\0 & 1 & 0\\\\0 & 0 & 3\\end{pmatrix}\\textcolor{blue}{\\begin{pmatrix}-1 & 2 / 3 & 0\\\\1 & -1 & 1\\\\0 & 1 / 3 & 0\\end{pmatrix}} \\\\\n c &= \\begin{pmatrix}1 & 2 & 0\\\\0 & 4 & 0\\\\0 & 1 & 1\\end{pmatrix} = \\textcolor{red}{\\begin{pmatrix}-1 & 0 & 2\\\\0 & 0 & 3\\\\1 & 1 & 1\\end{pmatrix}}\\begin{pmatrix}1 & 0 & 0\\\\0 & 1 & 0\\\\0 & 0 & 4\\end{pmatrix}\\textcolor{blue}{\\begin{pmatrix}-1 & 2 / 3 & 0\\\\1 & -1 & 1\\\\0 & 1 / 3 & 0\\end{pmatrix}} \\\\\n w &= \\begin{pmatrix}1 & 4 & 0\\\\0 & 7 & 0\\\\0 & 2 & 1\\end{pmatrix} = \\textcolor{red}{\\begin{pmatrix}-1 & 0 & 2\\\\0 & 0 & 3\\\\1 & 1 & 1\\end{pmatrix}}\\begin{pmatrix}1 & 0 & 0\\\\0 & 1 & 0\\\\0 & 0 & 7\\end{pmatrix}\\textcolor{blue}{\\begin{pmatrix}-1 & 2 / 3 & 0\\\\1 & -1 & 1\\\\0 & 1 / 3 & 0\\end{pmatrix}}\n \\end{align*}\n $$\n \n:::\n:::\n\n\nAll of these operators share the same eigenvectors, which can be seen\n in the columns of the left outer matrix and rows of the right outer one.\nThis means that composition of these operators only modifies the diagonal matrix,\n specifically the upper-left and lower-right eigenvalues.\n\nSome of these eigenvectors have special interpretations:\n\n- The left eigenvector (1, -1, 1) (right matrix, middle row) always has eigenvalue 1.\n - This means that the operation does not change the Euler characteristic.\n- The left eigenvector (3, -2, 0) (right matrix, top row), when applied to general polyhedra,\n corresponds to the edges and vertices added by the operation:\n - In *dk*, this vector has eigenvalue 0, forcing $3V = 2E$, i.e., all vertices to have degree 3.\n - In *c* and *w*, this vector has eigenvalue 1. These operators only add degree-3 vertices.\n\nThe (right) eigenvector (2, 3, 1), deserves special consideration.\nIt happens to be the same as the $F_6$-dependent component of *S*.\nIts eigenvalue is different for the three operators --\n 4 in the case of *c*, 3 in the case of *dk*, and 7 in the case of *w*.\nIn all three cases, these coincide with the norm of the GC parameters:\n\n$$\n\\begin{gather*}\n dkD = GC(1, 1) \\longrightarrow \\|1 + 1u\\| = 1^2 + 1 \\cdot 1 + 1^2 = 3 \\\\\n cD = GC(2, 0) \\longrightarrow \\|2 + 0u\\| = 2^2 + 2 \\cdot 0 + 0^2 = 4 \\\\\n wD = GC(2, 1) \\longrightarrow \\|2 + 1u\\| = 2^2 + 2 \\cdot 1 + 1^2 = 7\n\\end{gather*}\n$$\n\nCall this number $T = \\|a + bu\\| = a^2 + ab + b^2$.\nWe know that integers of this form are never congruent to 2 (mod 3).\nConveniently, this matches with the upper-left eigenvalue.\nAssuming for the sake of argument that this is true, it implies something interesting:\n **the GC operators produced from *T* are (combinatorially) closed under composition**.\nThis captures all powers of a given *T*, as well as products between it and other possible *T*s.\n\nHowever, this combinatorial view misses some of the picture, since some feature counts\n can be shared between two different classes at once.\nThis comes down to the existence of chiral pairs.\nFor example, [*ww*](https://levskaya.github.io/polyhedronisme/?recipe=K30wwD)\n = $(2, 1) \\circ (2, 1) = (5, 3)$,\n but\n [*ww'*](https://levskaya.github.io/polyhedronisme/?recipe=K30wrwD)\n = $(2, 1) \\circ (1, 2) = (7, 0)$.\n\nThis doesn't affect hexagon counts, so it means we can characterize the possible numbers given a certain *T*:\n\n$$\n\\begin{align*}\n g_T S &= g_T \\left(\n \\begin{pmatrix} 2 \\\\ 3 \\\\ 1 \\end{pmatrix} F_6\n + \\begin{pmatrix} 20 \\\\ 30 \\\\ 10 \\end{pmatrix}\n + \\begin{pmatrix} 0 \\\\ 0 \\\\ 2 \\end{pmatrix}\n \\right) \\\\\n &= T \\begin{pmatrix} 2 \\\\ 3 \\\\ 1 \\end{pmatrix} F_6\n + T \\begin{pmatrix} 20 \\\\ 30 \\\\ 10 \\end{pmatrix}\n + \\begin{pmatrix} 0 \\\\ 0 \\\\ 2 \\end{pmatrix} \\\\\n &= \\begin{pmatrix} T(2F_6 + 20) \\\\ T(3F_6 + 30) \\\\ T(F_6 + 10) + 2 \\end{pmatrix} \\\\[8pt]\n F'_6 &= T(F_6 + 10) - 10\n\\end{align*}\n$$\n\nThis demonstrates two things: the initial $F_6$ count from the choice of seed polygon *S* matters,\n and we can use valid values of *T* to produce a sequence of solutions.\n\n\nSearching for Other Seeds\n-------------------------\n\nThe dodecahedron, and by extension dodecahedral solutions, have icosahedral symmetry.\nTetrahedral symmetry, a sub-symmetry of this, was exploited in the previous post.\n\nHowever, there are further sub-symmetries which have not been encountered.\nFor example, some lesser symmetries of the dodecahedron are the vertices under rotation\n and inversion ([dihedral](https://en.wikipedia.org/wiki/Dihedral_group), degree 3)\n and the faces under rotation and inversion (dihedral, degree 5).\n\nRather than the typical construction based on paths between pentagons,\n this section will focus on certain base cases followed by a rudimentary application\n of the Conway operators above.\nThe particular symmetry group of each figure will be mentioned in each section.\n\n\n### Degenerate Deltahedra\n\n*Deltahedra* are polyhedra formed by equilateral triangles,\n a [table of which](https://en.wikipedia.org/wiki/Deltahedron) is available on Wikipedia.\nEquilateral triangles are important since three to five of them can be joined at a convex vertex,\n and if coplanar arrangements are allowed, up to six.\nDegree-5 and degree-6 vertices can be dualized or truncated to pentagons and hexagons respectively.\nAdditionally, when dualizing, all of the triangles become degree-3 vertices, producing a solution.\n\nAll convex deltahedra besides the tetrahedron and icosahedron contain degree-4 vertices.\nThese vertices pose a problem since their duals are quadrilaterals.\nHowever, it's reasonably easy to \"correct\" these vertices to a higher degree\n by selectively raising pyramids on faces (*augmenting*).\nThis section will focus on the coplanar entries on the table.\n\n\n#### Triangulated Rhombohedron ($Dih_3$)\n\n\nThe figure made by cutting the short diagonals of a\n [rhombohedron](https://en.wikipedia.org/wiki/Rhombohedron)\n is one such degree-4 vertex-free deltahedron.\nIt can also be seen as the figure formed by adding two triangular pyramids\n to opposite faces of an octahedron (a *biaugmentation*)\n or as a [*gyroelongated triangular bipyramid*](https://en.wikipedia.org/wiki/Gyroelongated_bipyramid).\n\nUnfortunately, the polyhedron viewer I typically use is unsuitable for visualizing this figure.\n[This viewer](https://drajmarsh.bitbucket.io/poly3d.html)\n supports more operations, and features an operator *z* which triangulates faces.\nIt produces the desired figure when applied to the cube\n (which is topologically equivalent to rhombohedra),\n\nTruncating the order-5 vertices produces an eggy-looking figure,\nIt inherits the dihedral symmetry of degree 3 from the vertices of the dodecahedron.\n\n:::: {.row layout-ncol=\"2\" layout-valign=\"center\"}\n::: {.column .slim-column layout=\"[[1,1],[1]]\"}\n![\n Biaugmented octahedron\n](./drajmarsh_zC.png)\n\n![\n Base case: order-5 truncation of left figure\n](./drajmarsh_t5zC.png)\n\n![\n Graph of the pole of the base case.
\n On the opposite pole, blue and red hexagons are exchanged.\n](./truncated gyroelongated bipyramid pole.png)\n:::\n\n::: {.column}\n\n::: {#8eb378d0 .cell .plain execution_count=3}\n\n::: {.cell-output .cell-output-display .cell-output-markdown execution_count=47}\nConway $F_6$ V E\n-------- ----------- ------ ------\n$F$ 6 32 48\n$F$ 38 96 144\n$F$ 54 128 192\n$F$ 102 224 336\n$F$ 134 288 432\n$F$ $16 T - 10$ $32 T$ $48 T$\n:::\n:::\n\n\n:::\n::::\n\n\n#### Base-Triangulated Frustum ($Dih_3$)\n\nConsider two faces of a figure, one a single equilateral triangle and the other\n containing four coplanar ones.\nThese two faces can be joined by three triples of triangles in the shape of a half-hexagon.\nThis figure can be seen as an octahedron with three tetrahedra placed around it (a *triaugmentation*),\n or as triangulation of the triangular prism.\n\nTruncating all vertices except those of degree 3 ($t_5 t_6$) produces a figure\n with 8 hexagons and indeed, 12 pentagons.\nUnfortunately, this figure is not easily constructible from normal seed polyhedra,\n even in the alternative viewer.\nHowever, it is still possible to operate on its projection as a planar graph:\n\n::: {#fig-triangular-frustum layout-ncol=\"4\"}\n![Initial graph](./triangulated_frustum.png)\n\n![After truncating degree-6 vertices](./t6_frustum.png)\n\n![After redrawing by \"inverting\" the interior and exterior](./inverted_t6_frustum.png)\n\n![After truncating degree-5 vertices](./t5_t6_frustum.png)\n:::\n\nThe pentagons gather in groups of 4 and are separated by three bands of hexagons,\n joined by two hexagons at either end.\nThe symmetry inherited by this figure is also dihedral of degree 3, but centered about a hexagon,\n rather than (a complex of) pentagons.\n\n::: {layout-ncol=\"2\" layout-valign=\"center\"}\n[\n![\n One-third of the figure, showing a group of four pentagons.
\n Pentagons in green, band hexagons in red, cap hexagons in blue.\n](./frustum_third.png){.column .slim-column}\n](./frustum_third.png){target=\"_blank_\"}\n\n::: {#9f849831 .cell .plain execution_count=4}\n\n::: {.cell-output .cell-output-display .cell-output-markdown execution_count=48}\nConway $F_6$ V E\n-------- ----------- ------ ------\n$F$ 8 36 54\n$F$ 44 108 162\n$F$ 62 144 216\n$F$ 116 252 378\n$F$ 152 324 486\n$F$ $18 T - 10$ $36 T$ $54 T$\n:::\n:::\n\n\n:::\n\n\n#### Others\n\nBefore their respective truncations, both of these figures of triangles which are compatible with\n (dual) Goldberg-Coxeter operators.\nFor example, larger triangular subdivisions (in the $u_n$ sense) will also give degenerate deltahedra,\nTriangular subdivision can add degree-6 vertices, so for the triangulated rhombohedron,\n an additional order-6 truncation needs to be done.\nDoing so means $t_5 t_6$ is applied to either case to form the solution figure.\n\nAn example subidivision is shown below.\nNote how the 3 pentagons at the pole have been separated from another 3 by a triangle of hexagons.\nI am fairly sure it is possible for mismatched pole configurations to be joined to one another\n for a more selective subdivision.\nOther Goldberg-Coxeter operations other than *u* can also be used;\n *dwd* for example connects non-polar pentagons with spirals of hexagons.\n\n::: {#fig-triangular-frustum-subdivision layout-ncol=\"2\"}\n![\n $t_5 t_6 uzC$
\n $F_6$ = 60, V = 140, E = 210\n](./drajmarsh_t5t6uzC.png)\n\n![\n $t_5 t_6 dwdzC$
\n $F_6$ = 114, V = 248, E = 372\n](./drajmarsh_t5t6dwdzC.png)\n:::\n\n\n### Truncated Trapezohedron ($Dih_6$)\n\nA *trapezohedron* is the polyhedral dual of an antiprism.\nAn example is a d10, which is the [pentagonal case](https://en.wikipedia.org/wiki/Pentagonal_trapezohedron).\nDespite the name \"*n*-gonal trapezohedron\", the figure is made entirely from 2*n* kites\n with the long edges meeting at points and the short edges meeting with each other.\n\nTruncating the order-*n* vertices where the long edges of the kites meet produces a\n [truncated trapezohedron](https://en.wikipedia.org/wiki/Truncated_trapezohedron),\nThese figures possess two regular *n*-gonal \"caps\" separated by a ring of pentagons.\nIn a way, they can be considered as a pentagonal analogue to prisms (quadrilaterals)\n and antiprisms (triangles).\n\nThe dodecahedron itself can be realized as a (order-5) truncated pentagonal trapezohedron ($t_5 d A_5$),\n which emphasizes its degree-5 dihedral symmetry.\nThe next largest truncated trapezohedron is the hexagonal case ($t_6 d A_6$),\n which contains and 2 hexagons and has dihedral symmetry of degree 6.\nThis is a symmetry beyond that of the icosahedron, but exists due to the symmetry of the hexagon.\n\n:::: {layout-ncol=\"2\" layout-valign=\"center\"}\n![\n Base case: truncated hexagonal trapezohedron\n](./polyhedronisme-K300t6dA6.png)\n\n::: {#ef876047 .cell .plain execution_count=5}\n\n::: {.cell-output .cell-output-display .cell-output-markdown execution_count=49}\nConway $F_6$ V E\n---------------------------------------------------------------------------------------------------- ----------- ------ ------\n[$T_6$](https://levskaya.github.io/polyhedronisme/?recipe=K300t6dA6){target=\"_blank\"} 2 24 36\n[$t_6dA_6 = T_6$](https://levskaya.github.io/polyhedronisme/?recipe=dkK300t6dA6){target=\"_blank\"} 26 72 108\n[$t_6dA_6 = T_6$](https://levskaya.github.io/polyhedronisme/?recipe=K300cK30t6dA6){target=\"_blank\"} 38 96 144\n[$t_6dA_6 = T_6$](https://levskaya.github.io/polyhedronisme/?recipe=K300wK30t6dA6){target=\"_blank\"} 74 168 252\n[$t_6dA_6 = T_6$](https://levskaya.github.io/polyhedronisme/?recipe=K300tkK30t6dA6){target=\"_blank\"} 98 216 324\n$t_6dA_6 = T_6$ $12 T - 10$ $24 T$ $36 T$\n:::\n:::\n\n\n::::\n\n\n### Medially-Separated Dodecahedron ($Dih_5$)\n\nAs the \"truncated pentagonal trapezohedron\", the dodecahedron can be separated into halves along an equator\n (specifically, along the [Petrie polygon](https://en.wikipedia.org/wiki/Petrie_polygon)).\nThis produces two halves with the pole configuration of the dodecahedron, joined together with hexagons.\nOrganizing the pentagons into two pairs of 6 is similar to the earlier case with the triangulated cube,\n but with a different arrangement of pentagons.\nThe resultant figure has dihedral symmetry of degree 5.\n\n:::: {layout-ncol=\"2\" layout-valign=\"center\"}\n::: {.column .slim-column layout=\"[[1,1],[1]]\"}\n![\n Dodecahedral graph\n](./dodecahedral graph.png)\n\n![\n Pole configuration of solution\n](./dodeca graph 2.png)\n\n![\n Complete base solution.
Note how the outer face has been rotated when compared with the dodecahedral graph.\n](./petrial expanded dodecahedron.png)\n:::\n\n::: {.column}\n\n::: {#23347dd4 .cell .plain execution_count=6}\n\n::: {.cell-output .cell-output-display .cell-output-markdown execution_count=50}\nConway $F_6$ V E\n-------- ----------- ------ ------\n$D_\\pi$ 5 30 45\n$D_\\pi$ 35 90 135\n$D_\\pi$ 50 120 180\n$D_\\pi$ 95 210 315\n$D_\\pi$ 125 270 405\n$D_\\pi$ $15 T - 10$ $30 T$ $45 T$\n:::\n:::\n\n\n:::\n::::\n\nInterestingly, this is also the first solution polyhedron with an odd number of faces.\n\n\n### Truncated Gyro-Pyramids ($Dih_5$, $Dih_6$)\n\n*g*, for \"gyro\", is a Conway operator I have used, but refused to explain.\nA *gyro-pyramid* (i.e., the gyro operator applied to pyramids,\n [example]](https://levskaya.github.io/polyhedronisme/?recipe=K300gY5))\n can roughly be described as a ring of 2*n* pentagons which alternate in orientation,\n the tips of which have an additional edge connecting to one of two antipodal endpoints.\nTruncating these endpoints these adds a face which is the same as the base of the original pyramid,\n surrounded by hexagons.\n\nSolution polyhedra can be found by examining the pentagonal ($t_5 g Y_5$) and hexagonal ($t_6 g Y_6$) cases.\nThe former has dihedral symmetry of degree 5 and the latter has that of degree 6.\n\n:::: {layout-ncol=\"2\" layout-valign=\"center\"}\n::: {.column .slim-column}\n![\n Base case 1: truncated pentagonal gyro-pyramid\n](./polyhedronisme-K300t5K30gY5.png){.column .slim-column}\n:::\n\n::: {#5873dced .cell .plain execution_count=7}\n\n::: {.cell-output .cell-output-display .cell-output-markdown execution_count=51}\nConway $F_6$ V E\n------------------------------------------------------------------------------------------------------ ----------- ------ ------\n[$G_5$](https://levskaya.github.io/polyhedronisme/?recipe=K300t5K30gY5){target=\"_blank\"} 10 40 60\n[$t_5gY_5 = G_5$](https://levskaya.github.io/polyhedronisme/?recipe=dkK300t5K30gY5){target=\"_blank\"} 50 120 180\n[$t_5gY_5 = G_5$](https://levskaya.github.io/polyhedronisme/?recipe=K300dudt5K30gY5){target=\"_blank\"} 70 160 240\n[$t_5gY_5 = G_5$](https://levskaya.github.io/polyhedronisme/?recipe=K300wK30t5K30gY5){target=\"_blank\"} 130 280 420\n[$t_5gY_5 = G_5$](https://levskaya.github.io/polyhedronisme/?recipe=tkK300t5K30gY5){target=\"_blank\"} 170 360 540\n$t_5gY_5 = G_5$ $20 T - 10$ $40 T$ $60 T$\n:::\n:::\n\n\n::: {.column .slim-column}\n![\n Base case 2: truncated hexagonal gyro-pyramid\n](./polyhedronisme-K300t6K30gY6.png)\n:::\n\n::: {#18e757b5 .cell .plain execution_count=8}\n\n::: {.cell-output .cell-output-display .cell-output-markdown execution_count=52}\nConway $F_6$ V E\n------------------------------------------------------------------------------------------------------ ----------- ------ ------\n[$G_6$](https://levskaya.github.io/polyhedronisme/?recipe=K300t6K30gY6){target=\"_blank\"} 14 48 72\n[$t_6gY_6 = G_6$](https://levskaya.github.io/polyhedronisme/?recipe=dkK300t6K30gY6){target=\"_blank\"} 62 144 216\n[$t_6gY_6 = G_6$](https://levskaya.github.io/polyhedronisme/?recipe=K300dudK40t6gY6){target=\"_blank\"} 86 192 288\n[$t_6gY_6 = G_6$](https://levskaya.github.io/polyhedronisme/?recipe=K300wK30t6K30gY6){target=\"_blank\"} 158 336 504\n[$t_6gY_6 = G_6$](https://levskaya.github.io/polyhedronisme/?recipe=tkK300t6K30gY6){target=\"_blank\"} 206 432 648\n$t_6gY_6 = G_6$ $24 T - 10$ $48 T$ $72 T$\n:::\n:::\n\n\n::::\n\nThe pentagonal case demonstrates something interesting: an appeal to the partition\n $12 = 10 + 1 + 1$.\nThe similar partition $12 = 5 + 5 + 1 + 1$ is unlikely to bear fruit,\n as an odd number of pentagons cannot alternate up and down.\n\n\n### Japanese Floor Tiling\n\n*Tatami* are a Japanese traditional style of floor mat, used even today in Japan\n as an intuitive measure for the surface area of living spaces.\nA single mat has an aspect ratio of 2:1, but the complexity comes in how they are arranged.\nLayouts are termed \"inauspicious\" (不祝儀敷き, *fushūgi-jiki*) when there are points where four mats meet,\n while \"auspicious\" (祝儀敷き, *shūgi-jiki*) layouts have all mats meet in threes.\nFor a sample of the fascination mathematicians have with these arrangements,\n you need only [search for \"tatami\" in the OEIS](http://oeis.org/search?q=tatami&language=english&go=Search)\n to find dozens of combinatorial sequences.\n\n::: {layout-ncol=\"3\"}\n![\n Inauspicious 4x4 layout\n](./tatami_inauspicious.png)\n\n![\n Auspicious 4x4 layout\n](./tatami_auspicious.png)\n\n![\n Possible topologies for a single mat\n](./tatami_topology.png)\n:::\n\nTopologically, each *tatami* mat in an arrangement can be thought of as either a quadrilateral,\n a pentagon or a hexagon.\nObviously, when considered alone, one mat is a rectangle, as is each mat in the inauspicious layout above.\nWhen two mats are affixed to one side of the mat, it becomes a pentagon,\n as in the auspicious layout above; all 8 mats are pentagons.\nWhen done to both sides, it becomes a hexagon.\n\nNaturally, the interplay between the latter two elements, as well as the condition\n that all mats meet in threes has a direct application to the problem at hand.\nIn fact, the graph of the whirl operation on the surface of a cube\n clearly makes the shape of one of these arrangements (with a half-mat included):\n\n:::: {#fig-tatami-polyhedron}\n::: {layout=\"[[-1,2,-1,2,-1]]\"}\n, via Wikimedia Commons\" href=\"https://commons.wikimedia.org/wiki/File:Conway_wC.png\">\n\"Conway\n\n\n, via Wikimedia Commons\" href=\"https://commons.wikimedia.org/wiki/File:Tearoom_layout.svg\">\n\"Tearoom\n\n:::\n\nImages were retrieved from Wikimedia and belong to their respective owners.\n::::\n\nTwo things pose a small issue for auspicious floor layouts:\n\n- when interpreted as a graph, the external face very quickly accumulates a large number of edges and\n- each of the mats at the corners of the room have a single degree-2 vertex\n\nBoth of these can be (partially) amended by connecting pairs of corner vertices,\n which creates additional faces while decreasing the number of edges on the perimeter.\n\nThe standard 4x6 auspicious layout is shown below, with corners connected and pentagons/hexagons identified.\nNext to it is an equivalent graph with a regular hexagon as an outer face.\nAs a 3D figure, it has only three hexagons and degree-3 dihedral symmetry, another odd number.\n\n:::: {layout-ncol=\"2\" layout-valign=\"center\"}\n::: {.column .slim-column .text-center}\n![](./tatami_graph.png){.image-wide}\n\n![](./regularized_tatami_graph.png){.image-wide}\n:::\n\n::: {#617ffa2d .cell .plain execution_count=9}\n\n::: {.cell-output .cell-output-display .cell-output-markdown execution_count=53}\nConway $F_6$ V E\n-------- ----------- ------ ------\n$M$ 3 26 39\n$M$ 29 78 117\n$M$ 42 104 156\n$M$ 81 182 273\n$M$ 107 234 351\n$M$ $13 T - 10$ $26 T$ $39 T$\n:::\n:::\n\n\n::::\n\nWhile *c* and *w* produce counts which are multiples of 3,\n *dk* somewhat strangely generates 29 and 107, which are both prime.\nIn fact, up to $(dk)^{11}$, the values are either prime or semiprime,\n and up to $(dk)^{28}$, the values are either 1-, 2-, or 3-, almost primes.\n\n\n#### Tatamified Projections\n\nWhile other rectangular *tatami* arrangements contain either too few or too many edges,\n graphs of solution polyhedra can be produced by using mats with nonstandard aspect ratios\n (and connecting corners as necessary).\n\nSome of these are shown below:\n\n::: {layout-ncol=\"2\"}\n![\n Dodecahedral graph, as a *tatami* arrangement with a single 4x1 mat.\n](./dodecahedron_tatami.png)\n\n![\n Medially-separated dodecahedron, as seen above.\n](./medial_dodecahedron.png)\n\n![\n Truncated icosahedral graph. Note the path of 2x1 mats.\n](./truncated_icosahedron_tatami.png)\n\n![\n Triacontahedron (not a base case shown above), initially similar to tI.\n](./18_hexagon_tatami.png)\n:::\n\n\nFinal Tabulation\n----------------\n\nThe solutions examined have been collected in the table below.\n\n::: {#428a6043 .cell .plain tbl-cap='T, T\\' are members of the sequence 1, 3, 4, 7, 9, 12, 13, 16, 19, 21... ([OEIS A003136](http://oeis.org/A003136))
\\\\*: This is the result of a GC operator applied to $2T - 14$, which requires *T* > 7' execution_count=10}\n\n::: {.cell-output .cell-output-display .cell-output-markdown execution_count=54}\nSymmetry Classification $F_6$ Example Values\n----------- ---------------------------------------------------------------- --------------------------------------------------- ---------------------------------------------\nIcosahedral Dodecahedral Goldberg $10 T - 10$ 20, 30, 60, 80, 110, 120, 150, 180, 200, 240\nTetrahedral Tetrahedral Goldberg Antitruncations $2 T - 14$ 4, 10, 12, 18, 24, 28, 36, 40, 42, 48\n GC(Antitruncations)* $T' \\left(2 T - 4\\right) - 10$ 32, 46, 50, 56, 70, 74, 78, 88, 92, 102\n Class I Edge-preserving $4 \\left(n + 1\\right)^{2} - 12$ 4, 24, 52, 88, 132, 184, 244, 312, 388, 472\n GC(Edge-preserves) $T' \\left(4 \\left(n + 1\\right)^{2} - 2\\right) - 10$ 4, 24, 32, 46, 52, 88, 92, 116, 126, 132\n$Dih_3$ Triangulated Rhombohedron $16 T - 10$ 6, 38, 54, 102, 134, 182, 198, 246, 294, 326\n Triangular Frustum $18 T - 10$ 8, 44, 62, 116, 152, 206, 224, 278, 332, 368\n [Tatamihedron](https://en.wikipedia.org/wiki/26-fullerene_graph) $13 T - 10$ 3, 29, 42, 81, 107, 146, 159, 198, 237, 263\n$Dih_5$ Medially-separated Dodecahedron $15 T - 10$ 5, 35, 50, 95, 125, 170, 185, 230, 275, 305\n Truncated Pentagonal Gyropyramid $20 T - 10$ 10, 50, 70, 130, 170, 230, 250, 310, 370, 410\n$Dih_6$ Truncated Hexagonal Trapezohedron $12 T - 10$ 2, 26, 38, 74, 98, 134, 146, 182, 218, 242\n Truncated Hexagonal Gyropyramid $24 T - 10$ 14, 62, 86, 158, 206, 278, 302, 374, 446, 494\n:::\n:::\n\n\nHere are a few hexagon counts from the table, sorted in ascending order:\n\n::: {#0f7ccb9d .cell execution_count=11}\n\n::: {.cell-output .cell-output-display .cell-output-markdown execution_count=55}\n2, 3, 4, 5, 6, 8, 10, 12, 14, 18, 20, 24, 26, 28, 29, 30, 32, 35, 36, 38...\n:::\n:::\n\n\nHowever, keep in mind that this list is *not* exhaustive.\nIn particular, it may be possible to construct additional entries by applying\n selective GC operators which preserve certain symmetries,\n like in the edge-preserving tetrahedral and deltahedron cases.\n\nSome small naturals which do not appear on the list are 1, 7, 9, 11, and 13.\nWithout constructing them, I am unsure whether they can exist.\nSome possible partitions to consider are $7 = 5 + 1 + 1$ (five equatorial and two polar hexagons, $Dih_5$)\n and $11 = 3 + 3 + 3 + 1 + 1$ (three hexagons along three lines of longitude\n or three triangles of hexagons pointing toward a pole, $Dih_3$).\n\n\nClosing\n-------\n\nDespite the Goldberg-Coxeter construction for dodecahedra being well-known,\n the 12 pentagons rule applies to a much broader class of polyhedra.\nIn fact, due to the GC construction, any polyhedron satisfying its conditions can be expanded\n and twisted into larger and larger solutions, which are far out of the way of ordinary soccer balls.\n\nPolyhedron images were generated using [polyHédronisme](https://levskaya.github.io/polyhedronisme/)\n and Dr. Andrew J. Marsh's [polyhedron generator](https://drajmarsh.bitbucket.io/poly3d.html).\nNets and graphs were created with GeoGebra.\nOther images were retrieved from Wikimedia and belong to their respective owners.\n\n", + "markdown": "---\ntitle: \"12 Pentagons, Part 3\"\ndescription: |\n Exploring even more symmetries based on the 12 pentagons condition.\nformat:\n html:\n html-math-method: katex\njupyter: python3\ndate: \"2021-06-09\"\ndate-modified: \"2025-03-16\"\ncategories:\n - geometry\n - combinatorics\n - symmetry\n---\n\n\n\n\n\nThis is the third part in an investigation into answering the following question:\n\n> A soccer ball is a (roughly spherical) figure made of pentagons and hexagons, each meeting 3 at a point.\n [T]here are 12 pentagons...how many hexagons can there be?\n\nThe [first post](../1) in the series proved the 12 pentagons portion and investigated\n (dodecahedral) Goldberg polyhedra.\nThe [second post](../2) investigated tetrahedral Goldberg polyhedra as well as other tetrahedral solutions.\nYet more unconventional solutions are presented below.\n\n\nCombinatorics of Goldberg-Coxeter Operators\n-------------------------------------------\n\nIn the first post, we established that a solution figure *S* has feature counts\n which are parametrized entirely by $F_6$:\n\n$$\nS = \\begin{pmatrix} v \\\\ e \\\\ f \\end{pmatrix}\n = \\begin{pmatrix}\n 2F_6 + 20 \\\\\n 3F_6 + 30 \\\\\n F_6 + 12\n \\end{pmatrix}\n = \\begin{pmatrix} 2 \\\\ 3 \\\\ 1 \\end{pmatrix} F_6\n + \\begin{pmatrix} 20 \\\\ 30 \\\\ 12 \\end{pmatrix}\n$$\n\nIn the last post, it was discovered that certain solutions cannot be created easily from seed polyhedra\n in a polyhedron viewer.\nThis prompts the question of how to calculate the vertex, edge, and face counts\n without acquiring them from the output of such a program.\nFortunatelly, each of the simple GC operators (*dk*, *c*, and *w*) has a combinatoric matrix form[^1].\n\n$$\n\\underset{\n \\text{Class I}\n}{\n c = \\begin{pmatrix}\n 1 & 2 & 0 \\\\\n 0 & 4 & 0 \\\\\n 0 & 1 & 1\n \\end{pmatrix}\n} \\quad\n\\underset{\n \\text{Class II}\n}{\n dk = \\begin{pmatrix}\n 0 & 2 & 0 \\\\\n 0 & 3 & 0 \\\\\n 1 & 0 & 1\n \\end{pmatrix}\n} \\quad\n\\underset{\n \\text{Class III}\n}{\n w = \\begin{pmatrix}\n 1 & 4 & 0 \\\\\n 0 & 7 & 0 \\\\\n 0 & 2 & 1\n \\end{pmatrix}\n}\n$$\n\n[^1]:\n These matrices can be derived by closely observing the vertex, edge, and face counts of a figure before\n and after applying the operator.\n Gather the feature counts of the tetrahedron, dodecahedron, and icosahedron into a matrix.\n\n $$\n \\begin{gather*}\n f \\begin{pmatrix} | & | & | \\\\ T & D & I \\\\ | & | & | \\end{pmatrix}\n = f \\begin{pmatrix} 4 & 20 & 12 \\\\ 6 & 30 & 30 \\\\ 4 & 12 & 20 \\end{pmatrix}\n = \\begin{pmatrix} | & | & | \\\\ fT & fD & fI \\\\ | & | & | \\end{pmatrix} \\\\\n f = \\begin{pmatrix} | & | & | \\\\ fT & fD & fI \\\\ | & | & | \\end{pmatrix}\n \\begin{pmatrix} 4 & 20 & 12 \\\\ 6 & 30 & 30 \\\\ 4 & 12 & 20 \\end{pmatrix}^{-1}\n \\end{gather*}\n $$\n\n This matrix is invertible, and the feature counts for *fT*, *fD*, and *fI* can be acquired from a viewer\n implementing the operator *f*.\n\nThese operators are labelled with the classes of Goldberg polyhedra they construct\n when applied to the dodecahedron.\n\n- *dkD* = *tI* = GC(1, 1)\n- *cD* = GC(2, 0)\n- *wD* = GC(2, 1)\n\nIt's worth noting again that *dk* alternates between producing Class I and Class II solutions.\n\n\n### Diagonalization\n\n*tk* is the square of the *dk* operator.\nUnlike it, it preserves solution class.\nPowers of a matrix are more readily expressed when the matrix is diagonalized.\nDiagonalizing each of these operators shows that they have something in common.\n\n::: {#305dbb2d .cell execution_count=2}\n\n::: {.cell-output .cell-output-display .cell-output-markdown execution_count=24}\n$$\n \\begin{align*}\n dk &= \\begin{pmatrix}0 & 2 & 0\\\\0 & 3 & 0\\\\1 & 0 & 1\\end{pmatrix} = \\textcolor{red}{\\begin{pmatrix}-1 & 0 & 2\\\\0 & 0 & 3\\\\1 & 1 & 1\\end{pmatrix}}\\begin{pmatrix}0 & 0 & 0\\\\0 & 1 & 0\\\\0 & 0 & 3\\end{pmatrix}\\textcolor{blue}{\\begin{pmatrix}-1 & 2 / 3 & 0\\\\1 & -1 & 1\\\\0 & 1 / 3 & 0\\end{pmatrix}} \\\\\n c &= \\begin{pmatrix}1 & 2 & 0\\\\0 & 4 & 0\\\\0 & 1 & 1\\end{pmatrix} = \\textcolor{red}{\\begin{pmatrix}-1 & 0 & 2\\\\0 & 0 & 3\\\\1 & 1 & 1\\end{pmatrix}}\\begin{pmatrix}1 & 0 & 0\\\\0 & 1 & 0\\\\0 & 0 & 4\\end{pmatrix}\\textcolor{blue}{\\begin{pmatrix}-1 & 2 / 3 & 0\\\\1 & -1 & 1\\\\0 & 1 / 3 & 0\\end{pmatrix}} \\\\\n w &= \\begin{pmatrix}1 & 4 & 0\\\\0 & 7 & 0\\\\0 & 2 & 1\\end{pmatrix} = \\textcolor{red}{\\begin{pmatrix}-1 & 0 & 2\\\\0 & 0 & 3\\\\1 & 1 & 1\\end{pmatrix}}\\begin{pmatrix}1 & 0 & 0\\\\0 & 1 & 0\\\\0 & 0 & 7\\end{pmatrix}\\textcolor{blue}{\\begin{pmatrix}-1 & 2 / 3 & 0\\\\1 & -1 & 1\\\\0 & 1 / 3 & 0\\end{pmatrix}}\n \\end{align*}\n $$\n \n:::\n:::\n\n\nAll of these operators share the same eigenvectors, which can be seen\n in the columns of the left outer matrix and rows of the right outer one.\nThis means that composition of these operators only modifies the diagonal matrix,\n specifically the upper-left and lower-right eigenvalues.\n\nSome of these eigenvectors have special interpretations:\n\n- The left eigenvector (1, -1, 1) (right matrix, middle row) always has eigenvalue 1.\n - This means that the operation does not change the Euler characteristic.\n- The left eigenvector (3, -2, 0) (right matrix, top row), when applied to general polyhedra,\n corresponds to the edges and vertices added by the operation:\n - In *dk*, this vector has eigenvalue 0, forcing $3V = 2E$, i.e., all vertices to have degree 3.\n - In *c* and *w*, this vector has eigenvalue 1. These operators only add degree-3 vertices.\n\nThe (right) eigenvector (2, 3, 1), deserves special consideration.\nIt happens to be the same as the $F_6$-dependent component of *S*.\nIts eigenvalue is different for the three operators --\n 4 in the case of *c*, 3 in the case of *dk*, and 7 in the case of *w*.\nIn all three cases, these coincide with the norm of the GC parameters:\n\n$$\n\\begin{gather*}\n dkD = GC(1, 1) \\longrightarrow \\|1 + 1u\\| = 1^2 + 1 \\cdot 1 + 1^2 = 3 \\\\\n cD = GC(2, 0) \\longrightarrow \\|2 + 0u\\| = 2^2 + 2 \\cdot 0 + 0^2 = 4 \\\\\n wD = GC(2, 1) \\longrightarrow \\|2 + 1u\\| = 2^2 + 2 \\cdot 1 + 1^2 = 7\n\\end{gather*}\n$$\n\nCall this number $T = \\|a + bu\\| = a^2 + ab + b^2$.\nWe know that integers of this form are never congruent to 2 (mod 3).\nConveniently, this matches with the upper-left eigenvalue.\nAssuming for the sake of argument that this is true, it implies something interesting:\n **the GC operators produced from *T* are (combinatorially) closed under composition**.\nThis captures all powers of a given *T*, as well as products between it and other possible *T*s.\n\nHowever, this combinatorial view misses some of the picture, since some feature counts\n can be shared between two different classes at once.\nThis comes down to the existence of chiral pairs.\nFor example, [*ww*](https://levskaya.github.io/polyhedronisme/?recipe=K30wwD)\n = $(2, 1) \\circ (2, 1) = (5, 3)$,\n but\n [*ww'*](https://levskaya.github.io/polyhedronisme/?recipe=K30wrwD)\n = $(2, 1) \\circ (1, 2) = (7, 0)$.\n\nThis doesn't affect hexagon counts, so it means we can characterize the possible numbers given a certain *T*:\n\n$$\n\\begin{align*}\n g_T S &= g_T \\left(\n \\begin{pmatrix} 2 \\\\ 3 \\\\ 1 \\end{pmatrix} F_6\n + \\begin{pmatrix} 20 \\\\ 30 \\\\ 10 \\end{pmatrix}\n + \\begin{pmatrix} 0 \\\\ 0 \\\\ 2 \\end{pmatrix}\n \\right) \\\\\n &= T \\begin{pmatrix} 2 \\\\ 3 \\\\ 1 \\end{pmatrix} F_6\n + T \\begin{pmatrix} 20 \\\\ 30 \\\\ 10 \\end{pmatrix}\n + \\begin{pmatrix} 0 \\\\ 0 \\\\ 2 \\end{pmatrix} \\\\\n &= \\begin{pmatrix} T(2F_6 + 20) \\\\ T(3F_6 + 30) \\\\ T(F_6 + 10) + 2 \\end{pmatrix} \\\\[8pt]\n F'_6 &= T(F_6 + 10) - 10\n\\end{align*}\n$$\n\nThis demonstrates two things: the initial $F_6$ count from the choice of seed polygon *S* matters,\n and we can use valid values of *T* to produce a sequence of solutions.\n\n\nSearching for Other Seeds\n-------------------------\n\nThe dodecahedron, and by extension dodecahedral solutions, have icosahedral symmetry.\nTetrahedral symmetry, a sub-symmetry of this, was exploited in the previous post.\n\nHowever, there are further sub-symmetries which have not been encountered.\nFor example, some lesser symmetries of the dodecahedron are the vertices under rotation\n and inversion ([dihedral](https://en.wikipedia.org/wiki/Dihedral_group), degree 3)\n and the faces under rotation and inversion (dihedral, degree 5).\n\nRather than the typical construction based on paths between pentagons,\n this section will focus on certain base cases followed by a rudimentary application\n of the Conway operators above.\nThe particular symmetry group of each figure will be mentioned in each section.\n\n\n### Degenerate Deltahedra\n\n*Deltahedra* are polyhedra formed by equilateral triangles,\n a [table of which](https://en.wikipedia.org/wiki/Deltahedron) is available on Wikipedia.\nEquilateral triangles are important since three to five of them can be joined at a convex vertex,\n and if coplanar arrangements are allowed, up to six.\nDegree-5 and degree-6 vertices can be dualized or truncated to pentagons and hexagons respectively.\nAdditionally, when dualizing, all of the triangles become degree-3 vertices, producing a solution.\n\nAll convex deltahedra besides the tetrahedron and icosahedron contain degree-4 vertices.\nThese vertices pose a problem since their duals are quadrilaterals.\nHowever, it's reasonably easy to \"correct\" these vertices to a higher degree\n by selectively raising pyramids on faces (*augmenting*).\nThis section will focus on the coplanar entries on the table.\n\n\n#### Triangulated Rhombohedron ($Dih_3$)\n\n\nThe figure made by cutting the short diagonals of a\n [rhombohedron](https://en.wikipedia.org/wiki/Rhombohedron)\n is one such degree-4 vertex-free deltahedron.\nIt can also be seen as the figure formed by adding two triangular pyramids\n to opposite faces of an octahedron (a *biaugmentation*)\n or as a [*gyroelongated triangular bipyramid*](https://en.wikipedia.org/wiki/Gyroelongated_bipyramid).\n\nUnfortunately, the polyhedron viewer I typically use is unsuitable for visualizing this figure.\n[This viewer](https://drajmarsh.bitbucket.io/poly3d.html)\n supports more operations, and features an operator *z* which triangulates faces.\nIt produces the desired figure when applied to the cube\n (which is topologically equivalent to rhombohedra),\n\nTruncating the order-5 vertices produces an eggy-looking figure,\nIt inherits the dihedral symmetry of degree 3 from the vertices of the dodecahedron.\n\n:::: {.row layout-ncol=\"2\" layout-valign=\"center\"}\n::: {.column .slim-column layout=\"[[1,1],[1]]\"}\n![\n Biaugmented octahedron\n](./drajmarsh_zC.png)\n\n![\n Base case: order-5 truncation of left figure\n](./drajmarsh_t5zC.png)\n\n![\n Graph of the pole of the base case.
\n On the opposite pole, blue and red hexagons are exchanged.\n](./truncated gyroelongated bipyramid pole.png)\n:::\n\n::: {.column}\n\n::: {#278918a0 .cell .plain execution_count=3}\n\n::: {.cell-output .cell-output-display .cell-output-markdown execution_count=25}\nConway $F_6$ V E\n-------- ----------- ------ ------\n$F$ 6 32 48\n$F$ 38 96 144\n$F$ 54 128 192\n$F$ 102 224 336\n$F$ 134 288 432\n$F$ $16 T - 10$ $32 T$ $48 T$\n:::\n:::\n\n\n:::\n::::\n\n\n#### Base-Triangulated Frustum ($Dih_3$)\n\nConsider two faces of a figure, one a single equilateral triangle and the other\n containing four coplanar ones.\nThese two faces can be joined by three triples of triangles in the shape of a half-hexagon.\nThis figure can be seen as an octahedron with three tetrahedra placed around it (a *triaugmentation*),\n or as triangulation of the triangular prism.\n\nTruncating all vertices except those of degree 3 ($t_5 t_6$) produces a figure\n with 8 hexagons and indeed, 12 pentagons.\nUnfortunately, this figure is not easily constructible from normal seed polyhedra,\n even in the alternative viewer.\nHowever, it is still possible to operate on its projection as a planar graph:\n\n::: {#fig-triangular-frustum layout-ncol=\"4\"}\n![Initial graph](./triangulated_frustum.png)\n\n![After truncating degree-6 vertices](./t6_frustum.png)\n\n![After redrawing by \"inverting\" the interior and exterior](./inverted_t6_frustum.png)\n\n![After truncating degree-5 vertices](./t5_t6_frustum.png)\n:::\n\nThe pentagons gather in groups of 4 and are separated by three bands of hexagons,\n joined by two hexagons at either end.\nThe symmetry inherited by this figure is also dihedral of degree 3, but centered about a hexagon,\n rather than (a complex of) pentagons.\n\n::: {layout-ncol=\"2\" layout-valign=\"center\"}\n[\n![\n One-third of the figure, showing a group of four pentagons.
\n Pentagons in green, band hexagons in red, cap hexagons in blue.\n](./frustum_third.png){.column .slim-column}\n](./frustum_third.png){target=\"_blank_\"}\n\n::: {#f2ba2049 .cell .plain execution_count=4}\n\n::: {.cell-output .cell-output-display .cell-output-markdown execution_count=26}\nConway $F_6$ V E\n-------- ----------- ------ ------\n$F$ 8 36 54\n$F$ 44 108 162\n$F$ 62 144 216\n$F$ 116 252 378\n$F$ 152 324 486\n$F$ $18 T - 10$ $36 T$ $54 T$\n:::\n:::\n\n\n:::\n\n\n#### Others\n\nBefore their respective truncations, both of these figures of triangles which are compatible with\n (dual) Goldberg-Coxeter operators.\nFor example, larger triangular subdivisions (in the $u_n$ sense) will also give degenerate deltahedra,\nTriangular subdivision can add degree-6 vertices, so for the triangulated rhombohedron,\n an additional order-6 truncation needs to be done.\nDoing so means $t_5 t_6$ is applied to either case to form the solution figure.\n\nAn example subidivision is shown below.\nNote how the 3 pentagons at the pole have been separated from another 3 by a triangle of hexagons.\nI am fairly sure it is possible for mismatched pole configurations to be joined to one another\n for a more selective subdivision.\nOther Goldberg-Coxeter operations other than *u* can also be used;\n *dwd* for example connects non-polar pentagons with spirals of hexagons.\n\n::: {#fig-triangular-frustum-subdivision layout-ncol=\"2\"}\n![\n $t_5 t_6 uzC$
\n $F_6$ = 60, V = 140, E = 210\n](./drajmarsh_t5t6uzC.png)\n\n![\n $t_5 t_6 dwdzC$
\n $F_6$ = 114, V = 248, E = 372\n](./drajmarsh_t5t6dwdzC.png)\n:::\n\n\n### Truncated Trapezohedron ($Dih_6$)\n\nA *trapezohedron* is the polyhedral dual of an antiprism.\nAn example is a d10, which is the [pentagonal case](https://en.wikipedia.org/wiki/Pentagonal_trapezohedron).\nDespite the name \"*n*-gonal trapezohedron\", the figure is made entirely from 2*n* kites\n with the long edges meeting at points and the short edges meeting with each other.\n\nTruncating the order-*n* vertices where the long edges of the kites meet produces a\n [truncated trapezohedron](https://en.wikipedia.org/wiki/Truncated_trapezohedron),\nThese figures possess two regular *n*-gonal \"caps\" separated by a ring of pentagons.\nIn a way, they can be considered as a pentagonal analogue to prisms (quadrilaterals)\n and antiprisms (triangles).\n\nThe dodecahedron itself can be realized as a (order-5) truncated pentagonal trapezohedron ($t_5 d A_5$),\n which emphasizes its degree-5 dihedral symmetry.\nThe next largest truncated trapezohedron is the hexagonal case ($t_6 d A_6$),\n which contains and 2 hexagons and has dihedral symmetry of degree 6.\nThis is a symmetry beyond that of the icosahedron, but exists due to the symmetry of the hexagon.\n\n:::: {layout-ncol=\"2\" layout-valign=\"center\"}\n![\n Base case: truncated hexagonal trapezohedron\n](./polyhedronisme-K300t6dA6.png)\n\n::: {#ff77c4b9 .cell .plain execution_count=5}\n\n::: {.cell-output .cell-output-display .cell-output-markdown execution_count=27}\nConway $F_6$ V E\n---------------------------------------------------------------------------------------------------- ----------- ------ ------\n[$T_6$](https://levskaya.github.io/polyhedronisme/?recipe=K300t6dA6){target=\"_blank\"} 2 24 36\n[$t_6dA_6 = T_6$](https://levskaya.github.io/polyhedronisme/?recipe=dkK300t6dA6){target=\"_blank\"} 26 72 108\n[$t_6dA_6 = T_6$](https://levskaya.github.io/polyhedronisme/?recipe=K300cK30t6dA6){target=\"_blank\"} 38 96 144\n[$t_6dA_6 = T_6$](https://levskaya.github.io/polyhedronisme/?recipe=K300wK30t6dA6){target=\"_blank\"} 74 168 252\n[$t_6dA_6 = T_6$](https://levskaya.github.io/polyhedronisme/?recipe=K300tkK30t6dA6){target=\"_blank\"} 98 216 324\n$t_6dA_6 = T_6$ $12 T - 10$ $24 T$ $36 T$\n:::\n:::\n\n\n::::\n\n\n### Medially-Separated Dodecahedron ($Dih_5$)\n\nAs the \"truncated pentagonal trapezohedron\", the dodecahedron can be separated into halves along an equator\n (specifically, along the [Petrie polygon](https://en.wikipedia.org/wiki/Petrie_polygon)).\nThis produces two halves with the pole configuration of the dodecahedron, joined together with hexagons.\nOrganizing the pentagons into two pairs of 6 is similar to the earlier case with the triangulated cube,\n but with a different arrangement of pentagons.\nThe resultant figure has dihedral symmetry of degree 5.\n\n:::: {layout-ncol=\"2\" layout-valign=\"center\"}\n::: {.column .slim-column layout=\"[[1,1],[1]]\"}\n![\n Dodecahedral graph\n](./dodecahedral graph.png)\n\n![\n Pole configuration of solution\n](./dodeca graph 2.png)\n\n![\n Complete base solution.
Note how the outer face has been rotated when compared with the dodecahedral graph.\n](./petrial expanded dodecahedron.png)\n:::\n\n::: {.column}\n\n::: {#21fa63d7 .cell .plain execution_count=6}\n\n::: {.cell-output .cell-output-display .cell-output-markdown execution_count=28}\nConway $F_6$ V E\n-------- ----------- ------ ------\n$D_\\pi$ 5 30 45\n$D_\\pi$ 35 90 135\n$D_\\pi$ 50 120 180\n$D_\\pi$ 95 210 315\n$D_\\pi$ 125 270 405\n$D_\\pi$ $15 T - 10$ $30 T$ $45 T$\n:::\n:::\n\n\n:::\n::::\n\nInterestingly, this is also the first solution polyhedron with an odd number of faces.\n\n\n### Truncated Gyro-Pyramids ($Dih_5$, $Dih_6$)\n\n*g*, for \"gyro\", is a Conway operator I have used, but refused to explain.\nA *gyro-pyramid* (i.e., the gyro operator applied to pyramids,\n [example]](https://levskaya.github.io/polyhedronisme/?recipe=K300gY5))\n can roughly be described as a ring of 2*n* pentagons which alternate in orientation,\n the tips of which have an additional edge connecting to one of two antipodal endpoints.\nTruncating these endpoints these adds a face which is the same as the base of the original pyramid,\n surrounded by hexagons.\n\nSolution polyhedra can be found by examining the pentagonal ($t_5 g Y_5$) and hexagonal ($t_6 g Y_6$) cases.\nThe former has dihedral symmetry of degree 5 and the latter has that of degree 6.\n\n:::: {layout-ncol=\"2\" layout-valign=\"center\"}\n::: {.column .slim-column}\n![\n Base case 1: truncated pentagonal gyro-pyramid\n](./polyhedronisme-K300t5K30gY5.png){.column .slim-column}\n:::\n\n::: {#5734093d .cell .plain execution_count=7}\n\n::: {.cell-output .cell-output-display .cell-output-markdown execution_count=29}\nConway $F_6$ V E\n------------------------------------------------------------------------------------------------------ ----------- ------ ------\n[$G_5$](https://levskaya.github.io/polyhedronisme/?recipe=K300t5K30gY5){target=\"_blank\"} 10 40 60\n[$t_5gY_5 = G_5$](https://levskaya.github.io/polyhedronisme/?recipe=dkK300t5K30gY5){target=\"_blank\"} 50 120 180\n[$t_5gY_5 = G_5$](https://levskaya.github.io/polyhedronisme/?recipe=K300dudt5K30gY5){target=\"_blank\"} 70 160 240\n[$t_5gY_5 = G_5$](https://levskaya.github.io/polyhedronisme/?recipe=K300wK30t5K30gY5){target=\"_blank\"} 130 280 420\n[$t_5gY_5 = G_5$](https://levskaya.github.io/polyhedronisme/?recipe=tkK300t5K30gY5){target=\"_blank\"} 170 360 540\n$t_5gY_5 = G_5$ $20 T - 10$ $40 T$ $60 T$\n:::\n:::\n\n\n::: {.column .slim-column}\n![\n Base case 2: truncated hexagonal gyro-pyramid\n](./polyhedronisme-K300t6K30gY6.png)\n:::\n\n::: {#d6e17e3e .cell .plain execution_count=8}\n\n::: {.cell-output .cell-output-display .cell-output-markdown execution_count=30}\nConway $F_6$ V E\n------------------------------------------------------------------------------------------------------ ----------- ------ ------\n[$G_6$](https://levskaya.github.io/polyhedronisme/?recipe=K300t6K30gY6){target=\"_blank\"} 14 48 72\n[$t_6gY_6 = G_6$](https://levskaya.github.io/polyhedronisme/?recipe=dkK300t6K30gY6){target=\"_blank\"} 62 144 216\n[$t_6gY_6 = G_6$](https://levskaya.github.io/polyhedronisme/?recipe=K300dudK40t6gY6){target=\"_blank\"} 86 192 288\n[$t_6gY_6 = G_6$](https://levskaya.github.io/polyhedronisme/?recipe=K300wK30t6K30gY6){target=\"_blank\"} 158 336 504\n[$t_6gY_6 = G_6$](https://levskaya.github.io/polyhedronisme/?recipe=tkK300t6K30gY6){target=\"_blank\"} 206 432 648\n$t_6gY_6 = G_6$ $24 T - 10$ $48 T$ $72 T$\n:::\n:::\n\n\n::::\n\nThe pentagonal case demonstrates something interesting: an appeal to the partition\n $12 = 10 + 1 + 1$.\nThe similar partition $12 = 5 + 5 + 1 + 1$ is unlikely to bear fruit,\n as an odd number of pentagons cannot alternate up and down.\n\n\n### Japanese Floor Tiling\n\n*Tatami* are a Japanese traditional style of floor mat, used even today in Japan\n as an intuitive measure for the surface area of living spaces.\nA single mat has an aspect ratio of 2:1, but the complexity comes in how they are arranged.\nLayouts are termed \"inauspicious\" (不祝儀敷き, *fushūgi-jiki*) when there are points where four mats meet,\n while \"auspicious\" (祝儀敷き, *shūgi-jiki*) layouts have all mats meet in threes.\nFor a sample of the fascination mathematicians have with these arrangements,\n you need only [search for \"tatami\" in the OEIS](http://oeis.org/search?q=tatami&language=english&go=Search)\n to find dozens of combinatorial sequences.\n\n::: {layout-ncol=\"3\"}\n![\n Inauspicious 4x4 layout\n](./tatami_inauspicious.png)\n\n![\n Auspicious 4x4 layout\n](./tatami_auspicious.png)\n\n![\n Possible topologies for a single mat\n](./tatami_topology.png)\n:::\n\nTopologically, each *tatami* mat in an arrangement can be thought of as either a quadrilateral,\n a pentagon or a hexagon.\nObviously, when considered alone, one mat is a rectangle, as is each mat in the inauspicious layout above.\nWhen two mats are affixed to one side of the mat, it becomes a pentagon,\n as in the auspicious layout above; all 8 mats are pentagons.\nWhen done to both sides, it becomes a hexagon.\n\nNaturally, the interplay between the latter two elements, as well as the condition\n that all mats meet in threes has a direct application to the problem at hand.\nIn fact, the graph of the whirl operation on the surface of a cube\n clearly makes the shape of one of these arrangements (with a half-mat included):\n\n:::: {#fig-tatami-polyhedron}\n::: {layout=\"[[-1,2,-1,2,-1]]\"}\n, via Wikimedia Commons\" href=\"https://commons.wikimedia.org/wiki/File:Conway_wC.png\">\n\"Conway\n\n\n, via Wikimedia Commons\" href=\"https://commons.wikimedia.org/wiki/File:Tearoom_layout.svg\">\n\"Tearoom\n\n:::\n\nImages were retrieved from Wikimedia and belong to their respective owners.\n::::\n\nTwo things pose a small issue for auspicious floor layouts:\n\n- when interpreted as a graph, the external face very quickly accumulates a large number of edges and\n- each of the mats at the corners of the room have a single degree-2 vertex\n\nBoth of these can be (partially) amended by connecting pairs of corner vertices,\n which creates additional faces while decreasing the number of edges on the perimeter.\n\nThe standard 4x6 auspicious layout is shown below, with corners connected and pentagons/hexagons identified.\nNext to it is an equivalent graph with a regular hexagon as an outer face.\nAs a 3D figure, it has only three hexagons and degree-3 dihedral symmetry, another odd number.\n\n:::: {layout-ncol=\"2\" layout-valign=\"center\"}\n::: {.column .slim-column .text-center}\n![](./tatami_graph.png){.image-wide}\n\n![](./regularized_tatami_graph.png){.image-wide}\n:::\n\n::: {#7ef1dfde .cell .plain execution_count=9}\n\n::: {.cell-output .cell-output-display .cell-output-markdown execution_count=31}\nConway $F_6$ V E\n-------- ----------- ------ ------\n$M$ 3 26 39\n$M$ 29 78 117\n$M$ 42 104 156\n$M$ 81 182 273\n$M$ 107 234 351\n$M$ $13 T - 10$ $26 T$ $39 T$\n:::\n:::\n\n\n::::\n\nWhile *c* and *w* produce counts which are multiples of 3,\n *dk* somewhat strangely generates 29 and 107, which are both prime.\nIn fact, up to $(dk)^{11}$, the values are either prime or semiprime,\n and up to $(dk)^{28}$, the values are either 1-, 2-, or 3-, almost primes.\n\n\n#### Tatamified Projections\n\nWhile other rectangular *tatami* arrangements contain either too few or too many edges,\n graphs of solution polyhedra can be produced by using mats with nonstandard aspect ratios\n (and connecting corners as necessary).\n\nSome of these are shown below:\n\n::: {layout-ncol=\"2\"}\n![\n Dodecahedral graph, as a *tatami* arrangement with a single 4x1 mat.\n](./dodecahedron_tatami.png)\n\n![\n Medially-separated dodecahedron, as seen above.\n](./medial_dodecahedron.png)\n\n![\n Truncated icosahedral graph. Note the path of 2x1 mats.\n](./truncated_icosahedron_tatami.png)\n\n![\n Triacontahedron (not a base case shown above), initially similar to tI.\n](./18_hexagon_tatami.png)\n:::\n\n\nFinal Tabulation\n----------------\n\nThe solutions examined have been collected in the table below.\n\n::: {#3cb5eec7 .cell .plain tbl-cap='T, T\\' are members of the sequence 1, 3, 4, 7, 9, 12, 13, 16, 19, 21... ([OEIS A003136](http://oeis.org/A003136))
\\\\*: This is the result of a GC operator applied to $2T - 14$, which requires *T* > 7' execution_count=10}\n\n::: {.cell-output .cell-output-display .cell-output-markdown execution_count=32}\nSymmetry Classification $F_6$ Example Values\n----------- ---------------------------------------------------------------- --------------------------------------------------- ---------------------------------------------\nIcosahedral Dodecahedral Goldberg $10 T - 10$ 20, 30, 60, 80, 110, 120, 150, 180, 200, 240\nTetrahedral Tetrahedral Goldberg Antitruncations $2 T - 14$ 4, 10, 12, 18, 24, 28, 36, 40, 42, 48\n GC(Antitruncations)* $T' \\left(2 T - 4\\right) - 10$ 32, 46, 50, 56, 70, 74, 78, 88, 92, 102\n Class I Edge-preserving $4 \\left(n + 1\\right)^{2} - 12$ 4, 24, 52, 88, 132, 184, 244, 312, 388, 472\n GC(Edge-preserves) $T' \\left(4 \\left(n + 1\\right)^{2} - 2\\right) - 10$ 32, 46, 88, 92, 116, 126, 158, 172, 176, 214\n$Dih_3$ Triangulated Rhombohedron $16 T - 10$ 6, 38, 54, 102, 134, 182, 198, 246, 294, 326\n Triangular Frustum $18 T - 10$ 8, 44, 62, 116, 152, 206, 224, 278, 332, 368\n [Tatamihedron](https://en.wikipedia.org/wiki/26-fullerene_graph) $13 T - 10$ 3, 29, 42, 81, 107, 146, 159, 198, 237, 263\n$Dih_5$ Medially-separated Dodecahedron $15 T - 10$ 5, 35, 50, 95, 125, 170, 185, 230, 275, 305\n Truncated Pentagonal Gyropyramid $20 T - 10$ 10, 50, 70, 130, 170, 230, 250, 310, 370, 410\n$Dih_6$ Truncated Hexagonal Trapezohedron $12 T - 10$ 2, 26, 38, 74, 98, 134, 146, 182, 218, 242\n Truncated Hexagonal Gyropyramid $24 T - 10$ 14, 62, 86, 158, 206, 278, 302, 374, 446, 494\n:::\n:::\n\n\nHere are a few hexagon counts from the table, sorted in ascending order:\n\n::: {#605a1db9 .cell execution_count=11}\n\n::: {.cell-output .cell-output-display .cell-output-markdown execution_count=33}\n2, 3, 4, 5, 6, 8, 10, 12, 14, 18, 20, 24, 26, 28, 29, 30, 32, 35, 36, 38...\n:::\n:::\n\n\nHowever, keep in mind that this list is *not* exhaustive.\nIn particular, it may be possible to construct additional entries by applying\n selective GC operators which preserve certain symmetries,\n like in the edge-preserving tetrahedral and deltahedron cases.\n\nSome small naturals which do not appear on the list are 1, 7, 9, 11, and 13.\nWithout constructing them, I am unsure whether they can exist.\nSome possible partitions to consider are $7 = 5 + 1 + 1$ (five equatorial and two polar hexagons, $Dih_5$)\n and $11 = 3 + 3 + 3 + 1 + 1$ (three hexagons along three lines of longitude\n or three triangles of hexagons pointing toward a pole, $Dih_3$).\n\n\nClosing\n-------\n\nDespite the Goldberg-Coxeter construction for dodecahedra being well-known,\n the 12 pentagons rule applies to a much broader class of polyhedra.\nIn fact, due to the GC construction, any polyhedron satisfying its conditions can be expanded\n and twisted into larger and larger solutions, which are far out of the way of ordinary soccer balls.\n\nPolyhedron images were generated using [polyHédronisme](https://levskaya.github.io/polyhedronisme/)\n and Dr. Andrew J. Marsh's [polyhedron generator](https://drajmarsh.bitbucket.io/poly3d.html).\nNets and graphs were created with GeoGebra.\nOther images were retrieved from Wikimedia and belong to their respective owners.\n\n", "supporting": [ "index_files" ], diff --git a/posts/pentagons/3/index.qmd b/posts/pentagons/3/index.qmd index f144700..52e6fa7 100644 --- a/posts/pentagons/3/index.qmd +++ b/posts/pentagons/3/index.qmd @@ -750,12 +750,13 @@ antitruncation = TotalHexagonRow( symmetry_group="Tetrahedral", class_name="Tetrahedral Goldberg Antitruncations", f6_expression=(2*t_param - 14), - t_filter=lambda x: x > 7, + t_filter=lambda t: t > 7, ) centered_edge = TotalHexagonRow( symmetry_group="Tetrahedral", class_name="Class I Edge-preserving", f6_expression=(4*((n_param + 1)**2 - 3)), + t_filter=lambda t: t > 1, ) def apply_gc(row: TotalHexagonRow, class_name: str) -> TotalHexagonRow: @@ -841,7 +842,7 @@ def get_example_counts(expr: sympy.Expr, amount: int, t_filter: TFilter = None): for t1 in t_params for t2 in t_params for n in range(1, 2*amount) - if t_filter is None or t_filter(t2) and t1 > 1 + if t_filter is None or t_filter(t2) and t1 > 2 ])) if count > 0 ][:amount]