Peter Kagey - cp's OEIS frontend

A383955 Decimal expansion of sqrt(5/3 - 2*sqrt(1/45)).

1, 1, 6, 9, 8, 3, 9, 4, 2, 0, 4, 6, 1, 9, 2, 5, 9, 2, 2, 6, 7, 5, 8, 0, 9, 6, 2, 2, 1, 4, 2, 8, 1, 1, 6, 1, 1, 3, 6, 1, 2, 7, 8, 0, 4, 3, 9, 7, 1, 5, 9, 2, 8, 5, 3, 0, 7, 7, 6, 7, 4, 3, 8, 2, 5, 8, 2, 9, 0, 1, 3, 5, 5, 2, 5, 3, 5, 2, 2, 4, 3, 3, 1, 6, 2, 0, 8

Offset: 1

Peter Kagey, Aug 19 2025

Let v_10 be a degree-10 vertex and v_3 be a degree-3 vertex of a triakis icosahedron centered at the origin. Then this is the ratio of norm(v_10)/norm(v_3).
The minimal polynomial is 45*x^4 - 150*x^2 + 121.
One choice of coordinates for the triakis icosahedron describes a degree-10 vertex of the triakis icosahedron as (0,1,(1+sqrt(5)/2)) and a degree-3 vertex as (5+7*sqrt(5)/22*(1, 1, 1).

			1.169839420461925922675809622142811611361278...

Wikipedia, Triakis_icosahedron

Cf. A378973, A378974, A378975, A378976, A378977, A380793, A380794, A382009.

RealDigits[Sqrt[(25 - 2*Sqrt[5])/15], 10, 100][[1]]

A385028 Number of face-connected components of polyhedral cells in the bisymmetric hendecahedral honeycomb up to translation, rotation, and reflection of the honeycomb.

Original entry on oeis.org

1, 1, 4, 16, 116, 903, 8551

Offset: 0

Peter Kagey, Aug 13 2025

These are "free polyforms" because they are counted up to rotation and reflection.

A bisymmetric hendecahedron is an 11-sided polyhedron that is similar to the convex hull of (-2,1,-1), (-2,1,1), (-1,-1,0), (0,-1,-1), (0,-1,1), (0,0,-2), (0,0,2), (0,2,0), (1,-1,0), (2,1,-1), and (2,1,1).

			For n = 2, the a(2) = 4 distinct compounds of two bisymmetric hendecahedra correspond to placing the four distinct types of faces (square, kite, rhombus, and triangle) together.

Peter Kagey, Illustration of the a(3)=16 components with three polyhedra.
Wikipedia, Bisymmetric hendecahedron
Jiangmei Wu and Guy Inchbald, Folding Space-Filling Bisymmetric Hendecahedron for a Large-Scale Art Installation, Bridges 2018 Conference Proceedings.
Jiangmei Wu, Synergia (sculpture made from bisymmetric hendecahedra).

Cf. A038119 (cubic), A038173 (rhombic dodecahedral), A038181 (truncated octahedral), A343909 (tetrahedral-octahedral), A365654 (square bipyramidal), A384254 (rectified cubic), A384274 (quarter cubic), A384754 (omnitruncated cubic), A385024 (tetragonal disphenoidal), A385025 (gyrobifastigium), A385026 (hexagonal prismatic), A385027 (triakis truncated tetrahedral), A385267 (half pyramidille), A385268 (oblate cubille), A385269 (quarter cubille), A385270 (elongated dodecahedral), A385271 (hexakis cubic), A385272 (phyllic disphenoidal), A385273 (quarter oblate octahedrille), A385274 (rhombic pyramidal), A385275 (square quarter pyramidille), A385276 (trapezo-rhombic dodecahedral), A385277 (triangular prismatic), A385278 (triangular pyramidille).

A385278 Number of face-connected components of polyhedral cells in the triangular pyramidille up to translation, rotation, and reflection of the honeycomb.

Original entry on oeis.org

1, 1, 3, 4, 16, 39, 152, 517, 2056, 8002, 32692, 134198, 561511, 2366909, 10075926, 43174057, 186208658, 807426463, 3518610508, 15400996653

Offset: 0

Peter Kagey and Bert Dobbelaere, Jun 25 2025

These are "free polyforms" because they are counted up to rotation and reflection.
The triangular pyramidille is dual to the cantitruncated cubic honeycomb.
The polyhedral cells are each 1/24 of a cube and are similar to the convex hull of (0,0,0), (2,0,0), (1,1,0), and (1,1,1).

Peter Kagey, Animation illustrating the a(5)=39 connected 5-cell components.
Wikipedia, Architectonic and catoptric tessellation
Wikipedia, Cubic honeycomb

Cf. A038119 (cubic), A038173 (rhombic dodecahedral), A038181 (truncated octahedral), A343909 (tetrahedral-octahedral), A365654 (square bipyramidal), A384254 (rectified cubic), A384274 (quarter cubic), A384754 (omnitruncated cubic), A385024 (tetragonal disphenoidal), A385025 (gyrobifastigium), A385026 (hexagonal prismatic), A385027 (triakis truncated tetrahedral), A385267 (half pyramidille), A385268 (oblate cubille), A385269 (quarter cubille), A385270 (elongated dodecahedral), A385271 (hexakis cubic), A385272 (phyllic disphenoidal), A385273 (quarter oblate octahedrille), A385274 (rhombic pyramidal), A385275 (square quarter pyramidille), A385276 (trapezo-rhombic dodecahedral), A385277 (triangular prismatic).

A385267 Number of face-connected components of half pyramidille cells in the half pyramidille up to translation, rotation, and reflection of the honeycomb.

Original entry on oeis.org

1, 1, 3, 4, 13, 29, 106, 331, 1222, 4390, 16588, 62865, 243217, 947711, 3732800, 14801687, 59103268, 237305379, 957738244, 3882631356, 15804400624

Offset: 0

Peter Kagey and Bert Dobbelaere, Jun 23 2025

The half pyramidille is the dual to the cantitruncated tetrahedral-octahedral honeycomb (also known as the runcicantic cubic honeycomb) which consists of truncated cuboctahedra, truncated cubes and truncated tetrahedra.

Each cell is similar to the convex hull of (0,0,0), (2,0,0), (2,2,0), and (1,1,1).

Peter Kagey, Animation illustrating the a(4) = 13 connected components consisting of 4 cells.
Wikipedia, Architectonic and catoptric tessellation
Wikipedia, Tetrahedral-octahedral honeycomb (see subsection "Runcicantic cubic honeycomb.")

Cf. A038119 (cubic), A038173 (rhombic dodecahedral), A038181 (truncated octahedral), A343909 (tetrahedral-octahedral), A365654 (square bipyramidal), A384254 (rectified cubic), A384274 (quarter cubic), A384754 (omnitruncated cubic), A385024 (tetragonal disphenoidal), A385025 (gyrobifastigium), A385026 (hexagonal prismatic), A385027 (triakis truncated tetrahedral), A385267 (half pyramidille), A385268 (oblate cubille), A385269 (quarter cubille), A385270 (elongated dodecahedral), A385271 (hexakis cubic), A385272 (phyllic disphenoidal), A385273 (quarter oblate octahedrille), A385274 (rhombic pyramidal), A385275 (square quarter pyramidille), A385276 (trapezo-rhombic dodecahedral), A385277 (triangular prismatic).

A385268 Number of face-connected components of oblate cubille cells in the oblate cubille up to translation, rotation, and reflection of the honeycomb.

Original entry on oeis.org

1, 1, 1, 3, 7, 24, 93, 427, 2043, 10412, 54072, 287121, 1543567, 8393603, 46040030, 254484780, 1415837030, 7922633039

Offset: 0

Peter Kagey and Bert Dobbelaere, Jun 23 2025

These are "free polyforms" because two connected components are considered to be the same if one can be rotated or reflected to match the other.
Each cell is 1/4 of a rhombic dodecahedron, and is similar to the convex hull of (-1,1,1), (0,0,0), (0,0,2), (0,2,0), (1,-1,1), (1,1,-1), (1,1,1), and (2,0,0).
The "oblate cubille" is also called the "trigonal trapezohedral honeycomb."

Peter Kagey, Animation illustrating the a(5)=24 connected 5-cell components.
Wikipedia, Architectonic and catoptric tessellation
Wikipedia, Trigonal trapezohedral honeycomb

Cf. A038119 (cubic), A038173 (rhombic dodecahedral), A038181 (truncated octahedral), A343909 (tetrahedral-octahedral), A365654 (square bipyramidal), A384254 (rectified cubic), A384274 (quarter cubic), A384754 (omnitruncated cubic), A385024 (tetragonal disphenoidal), A385025 (gyrobifastigium), A385026 (hexagonal prismatic), A385027 (triakis truncated tetrahedral), A385267 (half pyramidille), A385268 (oblate cubille), A385269 (quarter cubille), A385270 (elongated dodecahedral), A385271 (hexakis cubic), A385272 (phyllic disphenoidal), A385273 (quarter oblate octahedrille), A385274 (rhombic pyramidal), A385275 (square quarter pyramidille), A385276 (trapezo-rhombic dodecahedral), A385277 (triangular prismatic).

A385269 Number of face-connected components of quarter cubille cells in the quarter cubille up to translation, rotation, and reflection of the honeycomb.

Original entry on oeis.org

1, 1, 2, 4, 18, 67, 374, 2063, 12482, 76835, 486375, 3119695, 20275051, 133031450, 880300617, 5866722906

Offset: 0

Peter Kagey and Bert Dobbelaere, Jun 23 2025

These are "free polyforms" because they are counted up to rotation and reflection.
The quarter cubille is the dual to the runcic cubic honeycomb (equivalently, the cantellated tetrahedral-octahedral honeycomb) which consists of rhombicuboctahedra, cubes, and tetrahedra.
Each cell is similar to the convex hull of (0,2,2), (2,0,2), (2,2,0), (2,2,2), and (1,1,1).

Peter Kagey, Animation illustrating the a(4)=18 connected 4-cell components.
Wikipedia, Architectonic and catoptric tessellation
Wikipedia, Tetrahedral-octahedral honeycomb (see subsection "Quarter cubille.")

Cf. A038119 (cubic), A038173 (rhombic dodecahedral), A038181 (truncated octahedral), A343909 (tetrahedral-octahedral), A365654 (square bipyramidal), A384254 (rectified cubic), A384274 (quarter cubic), A384754 (omnitruncated cubic), A385024 (tetragonal disphenoidal), A385025 (gyrobifastigium), A385026 (hexagonal prismatic), A385027 (triakis truncated tetrahedral), A385267 (half pyramidille), A385268 (oblate cubille), A385269 (quarter cubille), A385270 (elongated dodecahedral), A385271 (hexakis cubic), A385272 (phyllic disphenoidal), A385273 (quarter oblate octahedrille), A385274 (rhombic pyramidal), A385275 (square quarter pyramidille), A385276 (trapezo-rhombic dodecahedral), A385277 (triangular prismatic).

A385270 Number of face-connected components of elongated dodecahedral cells in the elongated dodecahedral honeycomb up to translation, rotation, and reflection of the honeycomb.

Original entry on oeis.org

1, 1, 2, 8, 48, 362, 3530, 37861, 431383, 5059338, 60577228, 736054522, 9050344941, 112374575115

Offset: 0

Peter Kagey and Bert Dobbelaere, Jun 23 2025

These are "free polyforms" because they are counted up to rotation and reflection.

The symmetry group of the elongated dodecahedron is D_4h, which is prismatic symmetry of order 16.

Peter Kagey, Animation illustrating the a(4)=48 connected 4-cell components.
Wikipedia, Elongated dodecahedron
Wikipedia, Parallelohedron

Cf. A038119 (cubic), A038173 (rhombic dodecahedral), A038181 (truncated octahedral), A343909 (tetrahedral-octahedral), A365654 (square bipyramidal), A384254 (rectified cubic), A384274 (quarter cubic), A384754 (omnitruncated cubic), A385024 (tetragonal disphenoidal), A385025 (gyrobifastigium), A385026 (hexagonal prismatic), A385027 (triakis truncated tetrahedral), A385267 (half pyramidille), A385268 (oblate cubille), A385269 (quarter cubille), A385270 (elongated dodecahedral), A385271 (hexakis cubic), A385272 (phyllic disphenoidal), A385273 (quarter oblate octahedrille), A385274 (rhombic pyramidal), A385275 (square quarter pyramidille), A385276 (trapezo-rhombic dodecahedral), A385277 (triangular prismatic).

A385271 Number of face-connected components of square pyramidal cells in the hexakis cubic honeycomb up to translation, rotation, and reflection of the honeycomb.

Original entry on oeis.org

1, 1, 2, 3, 9, 17, 60, 166, 606, 2106, 8046, 30801, 122442, 491539, 2007571, 8272122, 34408439, 144084776, 607112043, 2571118048, 10938419260, 46720437135

Offset: 0

Peter Kagey and Bert Dobbelaere, Jun 25 2025

These are "free polyforms" because they are counted up to rotation and reflection.
The "hexakis cubic honeycomb" is also called the "pyramidille" and is dual to the truncated cubic honeycomb.
The square pyramidal cells are similar to the convex hull of (0,0,0), (2,0,0), (0,2,0) (2,2,0), and (1,1,1).

Peter Kagey, Animation illustrating the a(5)=17 connected 5-cell components.
Wikipedia, Architectonic and catoptric tessellation
Wikipedia, Tetragonal disphenoid honeycomb, see section "Hexakis cubic honeycomb."

Cf. A038119 (cubic), A038173 (rhombic dodecahedral), A038181 (truncated octahedral), A343909 (tetrahedral-octahedral), A365654 (square bipyramidal), A384254 (rectified cubic), A384274 (quarter cubic), A384754 (omnitruncated cubic), A385024 (tetragonal disphenoidal), A385025 (gyrobifastigium), A385026 (hexagonal prismatic), A385027 (triakis truncated tetrahedral), A385267 (half pyramidille), A385268 (oblate cubille), A385269 (quarter cubille), A385270 (elongated dodecahedral), A385271 (hexakis cubic), A385272 (phyllic disphenoidal), A385273 (quarter oblate octahedrille), A385274 (rhombic pyramidal), A385275 (square quarter pyramidille), A385276 (trapezo-rhombic dodecahedral), A385277 (triangular prismatic).

A385272 Number of face-connected components of phyllic disphenoidal cells in the phyllic disphenoidal honeycomb up to translation, rotation, and reflection of the honeycomb.

Original entry on oeis.org

1, 1, 2, 4, 13, 38, 141, 515, 2043, 8176, 33706, 140471, 593705, 2531933, 10893811, 47202599, 205843902, 902644191, 3977976135, 17609163491

Offset: 0

Peter Kagey and Bert Dobbelaere, Jun 25 2025

These are "free polyforms" because they are counted up to rotation and reflection.
The "phyllic disphenoidal honeycomb" is also called the "eighth pyramidille," and its dual is the omnitruncated cubic honeycomb.
The phyllic disphenoidal cells are similar to the convex hull of (0,0,0), (1,0,0), (1,1,0), and (1,1,1).

Peter Kagey, Animation illustrating the a(5)=38 connected 5-cell components.
Wikipedia, Architectonic and catoptric tessellation
Wikipedia, Tetragonal disphenoid honeycomb, see section "Phyllic disphenoidal honeycomb."

Cf. A038119 (cubic), A038173 (rhombic dodecahedral), A038181 (truncated octahedral), A343909 (tetrahedral-octahedral), A365654 (square bipyramidal), A384254 (rectified cubic), A384274 (quarter cubic), A384754 (omnitruncated cubic), A385024 (tetragonal disphenoidal), A385025 (gyrobifastigium), A385026 (hexagonal prismatic), A385027 (triakis truncated tetrahedral), A385267 (half pyramidille), A385268 (oblate cubille), A385269 (quarter cubille), A385270 (elongated dodecahedral), A385271 (hexakis cubic), A385272 (phyllic disphenoidal), A385273 (quarter oblate octahedrille), A385274 (rhombic pyramidal), A385275 (square quarter pyramidille), A385276 (trapezo-rhombic dodecahedral), A385277 (triangular prismatic).

A385273 Number of face-connected components of polyhedral cells in the quarter oblate octahedrille up to translation, rotation, and reflection of the honeycomb.

Original entry on oeis.org

1, 1, 2, 6, 24, 114, 647, 3883, 24605, 159837, 1060450, 7137627, 48624639, 334475495, 2319909330, 16205238283

Offset: 0

Peter Kagey and Bert Dobbelaere, Jun 25 2025

These are "free polyforms" because they are counted up to rotation and reflection.
The quarter oblate octahedrille is dual to the cantellated cubic honeycomb.
The cells of the quarter oblate octahedrille are similar to the convex hull of (0,0,0), (1,0,0), (0, 1, 0), (1,1,1), and (1,1,-1).

Peter Kagey, Animation illustrating the a(4)=24 connected 4-cell components.
Wikipedia, Architectonic and catoptric tessellation

Cf. A038119 (cubic), A038173 (rhombic dodecahedral), A038181 (truncated octahedral), A343909 (tetrahedral-octahedral), A365654 (square bipyramidal), A384254 (rectified cubic), A384274 (quarter cubic), A384754 (omnitruncated cubic), A385024 (tetragonal disphenoidal), A385025 (gyrobifastigium), A385026 (hexagonal prismatic), A385027 (triakis truncated tetrahedral), A385267 (half pyramidille), A385268 (oblate cubille), A385269 (quarter cubille), A385270 (elongated dodecahedral), A385271 (hexakis cubic), A385272 (phyllic disphenoidal), A385273 (quarter oblate octahedrille), A385274 (rhombic pyramidal), A385275 (square quarter pyramidille), A385276 (trapezo-rhombic dodecahedral), A385277 (triangular prismatic).

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User: Peter Kagey

A383955 Decimal expansion of sqrt(5/3 - 2*sqrt(1/45)).

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A385028 Number of face-connected components of polyhedral cells in the bisymmetric hendecahedral honeycomb up to translation, rotation, and reflection of the honeycomb.

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A385278 Number of face-connected components of polyhedral cells in the triangular pyramidille up to translation, rotation, and reflection of the honeycomb.

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A385267 Number of face-connected components of half pyramidille cells in the half pyramidille up to translation, rotation, and reflection of the honeycomb.

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A385268 Number of face-connected components of oblate cubille cells in the oblate cubille up to translation, rotation, and reflection of the honeycomb.

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A385269 Number of face-connected components of quarter cubille cells in the quarter cubille up to translation, rotation, and reflection of the honeycomb.

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A385270 Number of face-connected components of elongated dodecahedral cells in the elongated dodecahedral honeycomb up to translation, rotation, and reflection of the honeycomb.

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A385271 Number of face-connected components of square pyramidal cells in the hexakis cubic honeycomb up to translation, rotation, and reflection of the honeycomb.

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A385272 Number of face-connected components of phyllic disphenoidal cells in the phyllic disphenoidal honeycomb up to translation, rotation, and reflection of the honeycomb.

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A385273 Number of face-connected components of polyhedral cells in the quarter oblate octahedrille up to translation, rotation, and reflection of the honeycomb.

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