User:Peter Kagey

I graduated from Oregon State University in June 2014 with a B.S. in Mathematics. From December 2014 until August 2017, I worked as a back end web engineer at a health technology company. From August 2017–July 2022, I was in a Ph.D. program at the University of Southern California. From 2022–2024, I was visiting assistant professor of mathematics at Harvey Mudd College. As of 2025, I am an assistant professor in the department of Mathematics and Statistics at Cal Poly Pomona.

Most of the scripts that I have written to generate OEIS sequences can be found on my OEIS Github Repository. I work mostly in Haskell.

Feel free to contact me at [email protected] or on LinkedIn.

Tilings of the $n\times m$ grid, cylinder, and torus

Tilings of the $n\times n$ grid

	$⟨r,f⟩\cong D_8$	$⟨r^2,rf⟩\cong V$	$⟨r⟩\cong C_4$	$⟨rf⟩\cong C_2$
$⟨r,f⟩$	A054247	—	—	—
$⟨r^2,f⟩$	A367522*	—	—	—
$⟨r^2,rf⟩$	A295229*	A367526*	—	—
$⟨r⟩$	A367523*	—	A047937	—
$⟨f⟩$	A367524*	—	—	—
$⟨rf⟩$	A302484	A367527*	—	A200564
$⟨r^2⟩$	A367524*	A367528*	A367531*	—
$⟨⟩$	A367525*	A367529*	A367532*	A103488

Tilings of the $n\times n$ torus

	$⟨r,f⟩\cong D_8$	$⟨r^2,rf⟩\cong V$	$⟨r⟩\cong C_4$	$⟨rf⟩\cong C_2$	$⟨⟩$
$⟨r,f⟩$	A255016	—	—	—	—
$⟨r^2,f⟩$	A367533*	—	—	—	—
$⟨r^2,rf⟩$	A295223*	A368139*	—	—	—
$⟨r⟩$	A367534*	—	A368143*	—	—
$⟨f⟩$	A367535*	—	—	—	—
$⟨rf⟩$	A367536*	A368140*	—	A255015	—
$⟨r^2⟩$	A368137*	A368141*	A368144*	—	—
$⟨⟩$	A368138*	A368142*	A368145*	A367530*	A179043

Tilings of the $n\times m$ grid

	$⟨r^2,f⟩$	$⟨f⟩$	$⟨r^2⟩$	$⟨⟩$
$⟨r^2,f⟩$	A225910	—	—	—
$⟨f⟩$	A368218*	A368221*	—	—
$⟨r^2⟩$	A368219*	—	A368223*	—
$⟨⟩$	A368220*	A368222*	A368224*	A117401

Tilings of the $n\times m$ cylinder

	$⟨r^2,f⟩$	$⟨f⟩$	$⟨r^{2}f⟩$	$⟨r^2⟩$	$⟨⟩$
$⟨r^2,f⟩$	A368253*	—	—	—	—
$⟨f⟩$	A368254*	A368258*	—	—	—
$⟨r^{2}f⟩$	A368255*	—	A368260*	—	—
$⟨r^2⟩$	A368256*	—	—	A368262*	—
$⟨⟩$	A368257*	A368259*	A368261*	A368263*	A368264*

Tilings of the $n\times m$ torus

	$⟨r^2,f⟩$	$⟨f⟩$	$⟨r^2⟩$	$⟨⟩$
$⟨r^2,f⟩$	A222188	—	—	—
$⟨f⟩$	A368302*	A368305*	—	—
$⟨r^2⟩$	A368303*	—	A368307*	—
$⟨⟩$	A368304*	A368306*	A3683088	A184271

Coloring polyhedra

Platonic solids rotation/reflection

	Vertices	Edges	Faces
Tetrahedron	A000332	A063842	A000332
Octahedron	A198833	A199406	A128766
Cube	A128766	A199406	A198833
Dodecahedron	A252704	A337963	A252705
Icosahedron	A252705	A337963	A252704

Archimedean/Catalan solids with full symmetry

	V / F	Edges	F / V
Truncated tetrahedron / triakis tetrahedron	A060530	?	?
Cuboctahedron / Rhombic dodecahedron	A199406	?	A274901
Truncated cube / Triakis octahedron	A274900	?	A274901
Truncated octahedron / Tetrakis hexahedron	A378473	?	A274901
Rhombicuboctahedron / Deltoidal icositetrahedron	A274900	?	?
Truncated cuboctahedron / disdyakis dodecahedron	A378474	?	?
Snub cube / pentagonal icositetrahedron	A378475	?	?
Icosidodecahedron / rhombic triacontahedron	A337963	?	?
Truncated dodecahedron / triakis icosahedron	A378476	?	?
Truncated icosahedron / pentakis dodecahedron	A378476	?	?
Rhombicosidodecahedron / deltoidal hexecontahedron	A378476	?	?
Truncated icosidodecahedron / disdyakis triacontahedron	A378477	?	?
Snub dodecahedron / pentagonal hexecontahedron	A378478	?	?

Polytopes in lattices

Polygons in lattices
	2D					3D
	Square	Rectangular	Centered Square	Triangular	Centered Hexagonal	Cubic	Tetrahedral	Octahedral
Equilateral Triangle	-	-	-	A000332	A008893	A102698	A334581*	?
Square	A002415	A130684	A006324	-	-	A334881*	A334891*	?
Regular Hexagon	-	-	-	A011779 (?)	A000537	A338322*	?	?
Regular Polygon	A002415	A130684	A006324	?	A339483*	A338323*	?	?
Triangle	A045996	A334705	?	?	A241223	?	?	?
Rectangle	A085582	A289832	?	-	-	?	?	?
Right Triangle	A077435	?	?	?	A241225	?	?	?

Other polygons in lattices
	Square	Rectangular	Centered Hexagonal	Centered Square	Triangular
Acute Triangle	A190019		A241224
Obtuse Triangle	A190020		A241226
Scalene Triangle	A190312		A241227
Isosceles Triangle	A186434	A271910	A241228
Acute Isosceles	A190317	A272625	A241229
Obtuse Isosceles	A190318	A272624	A241230
Concave Quadrilateral	A189412
Convex Quadrilateral	A189413
Quadrilateral	A189414
Trapezoid	A189415
Parallelogram	A189416
Kite	A189417
Rhombus	A189418

Polyhedra in lattices
	Cubic	Tetrahedral	Octahedral
Regular Tetrahedron	A103158	A269747	?
Cube	A098928	?	?
Octahedron	A178797	?	?
Platonic Solid	A338791	?	?

Polyforms on polyhedra

Symmetry	Polyhedron	One-sided	Free
Tetrahedral	Tetrahedron	1,1,1,1,1	1,1,1,1,1
	Triakis tetrahedron	A383826	A383825
	Tetrakis hexahedron		A383827
Octahedral	Cube	1,1,1,2,2,1,1	1,1,1,2,2,1,1
	Octahedron	1,1,1,1,4,2,3,1,1	A369605 (row 3)
	Rhombic dodecahedron	A383799	A333333 (row 3)
	Triakis octahedron	A383801	A383800
	Tetrakis hexahedron	A383803	A383802
	Deltoidal icositetrahedron	A383805	A383804
	Disdyakis dodecahedron	A383807	A383806
	Pentagonal icositetrahedron	A383808	-
Icosahedral	Dodecahedron	A030137	A030135
	Icosahedron	A030138	A030136
	Rhombic triacontahedron	A383491	A383490
	Triakis icosahedron	A383493	A383492
	Pentakis dodecahedron	A383495	A383494
	Deltoidal hexecontahedron	A340635	A383804
	Disdyakis triacontahedron	A383497	A383496
	Pentagonal hexecontahedron	A383786	-

Polyforms on tilings

Family	Tiling	Sequence	Link
Regular (Euclidean)	Square ({4,4})	A000105	[1]
	Hexagonal ({6,3})	A000228	[2]
	Triangular ({3,6})	A000577	[3]
Regular (hyperbolic)	{3,7} (hyperbolic)	A330659	[4]
	{3,∞} (hyperbolic)	A000207	[5]
	{4,5} (hyperbolic)	A119611	[6]
	{4,6} (hyperbolic)	A332930	[7]
	{4,∞} (hyperbolic)	A005036	[8]
	{5,4} (hyperbolic)	A330211	[9]
	{7,3} (hyperbolic)	A333018	[10]
	{k,∞} (hyperbolic)	A295260	[11]
Higher Dimensions	Cubic	A038119	[12]
	Tesseractic	A068870	[13]
	Hypercubic (k-D)	A330891	[14]
	Hypercubic (∞-D)	A005519
	Alternated cubic	A343909	[15]
	Bitruncated cubic	A038181	[16]
	Rhombic dodecahedral	A038173	[17]

Uniform Tiling	Sequence	Link	Dual tiling	Sequence	Link
Snub Square	A309159	[18]	Cairo pentagonal tiling	A159866	[19]
Truncated square	A343577	[20]	Tetrakis square tiling	A197465	[21]
Elongated triangular	A345076	[22]	Prismatic pentagonal tiling	A197156	[23]
Truncated hexagonal	A343406	[24]	Triakis triangular tiling	A197465	[25]
Trihexagonal tiling	A343398	[26]	Rhombille tiling	A197459	[27]
Rhombitrihexagonal	A344211	[28]	Deltoidal trihexagonal tiling	A057786	[29]
Truncated trihexagonal	A344213	[30]	Kishombille tiling	A197462	[31]
Snub trihexagonal	A383908	[32]	Floret pentagonal tiling	A197159	[33]