Regular triangular tilings of the torus

Regular triangular tilings of the torus have traditional subscripted labels with the format "(a,b)", with a≤b and a+b even. The number of triangles in a tiling is (a²+3b²)/2.

A cleaner way of labelling the tilings uses labels with the format "(i,j)", with their order irrelevant. The number of triangles in a tiling is 2(i²+ij+j²). This labelling system is based on Eisenstein integers, which reflect the sixfold symmetry of these tilings. To convert between the traditional and Eisenstein labels, use
i=(b-a)/2,  j=(b+a)/2      and      a=abs(i-j),  b=i+j.

Name	Schläfli	Eisenstein i,j	V / F / E	mV, mF	notes	C&D no.	images
{3,6}(1,1)	{3,6}2	0,1	1 / 2 / 3	6,3	α δ' ξ'	R1.t1-1	2
{3,6}(0,2)	{3,6}6	1,1	3 / 6 / 9	3,1	ο' ο°'	R1.t0-2	3
{3,6}(2,2)	{3,6}4	0,2	4 / 8 / 12	2,1	ξ'	R1.t2-2	1
the dual Heawood map	{3,6}14	1,2	7 / 14 / 21	1,1		C1.t1-3	1
{3,6}(3,3)	{3,6}6	0,3	9 / 18 / 27	1,1	ξ' ξ°'	R1.t3-3	1
{3,6}(0,4)	{3,6}12	2,2	12 / 24 / 36	1,1	ο'	R1.t0-4	1
{3,6}(2,4)	{3,6}26	1,3	13 / 26 / 39	1,1		C1.t2-4	1
{3,6}(4,4)	{3,6}8	0,4	16 / 32 / 48	1,1	ξ'	R1.t4-4	1
{3,6}(1,5)	{3,6}38	2,3	19 / 38 / 57	1,1		C1.t1-5	1
{3,6}(3,5)	{3,6}42	1,4	21 / 42 / 63	1,1		C1.t3-5	1
{3,6}(5,5)	{3,6}10	0,5	25 / 50 / 75	1,1	ξ'	R1.t5-5	1
{3,6}(0,6)	{3,6}18	3,3	27 / 54 / 81	1,1	ο'	R1.t0-6	1
{3,6}(2,6)	{3,6}28	2,4	28 / 56 / 84	1,1		C1.t2-6	1
{3,6}(4,6)	{3,6}62	1,5	31 / 62 / 93	1,1		C1.t4-6	1
{3,6}(6,6)	{3,6}12	0,6	36 / 72 / 108	1,1	ξ'	R1.t6-6	1
{3,6}(1,7)	{3,6}74	3,4	37 / 74 / 111	1,1		C1.t1-7	1
{3,6}(3,7)	{3,6}78	2,5	39 / 78 / 117	1,1		C1.t3-7	1
{3,6}(5,7)	{3,6}86	1,6	43 / 86 / 129	1,1		C1.t5-7	1
{3,6}(0,8)	{3,6}24	4,4	48 / 96 / 144	1,1	ο'	R1.t0-8	1
{3,6}(7,7)	{3,6}14	0,7	49 / 98 / 147	1,1	ξ'	R1.t7-7	1
{3,6}(2,8)	{3,6}98	3,5	49 / 98 / 147	1,1		C1.t2-8	1
{3,6}(8,8)	{3,6}16	0,8	64 / 128 / 192	1,1	ξ'	R1.t8-8	(see ser ξ')
{3,6}(0,10)	{3,6}30	5,5	75 / 150 / 225	1,1	ο'	R1.t0-10	(see ser ο')
{3,6}(9,9)	{3,6}18	0,9	81 / 162 / 243	1,1	ξ'	R1.t9-9	(see ser ξ')
{3,6}(10,10)	{3,6}20	0,10	100 / 200 / 300	1,1	ξ'	R1.t10-10	(see ser ξ')
{3,6}(0,12)	{3,6}36	6,6	108 / 216 / 324	1,1	ο'	R1.t0-12	(see ser ο')
{3,6}(11,11)	{3,6}22	0,11	121 / 242 / 363	1,1	ξ'	R1.t11-11	(see ser ξ')
{3,6}(12,12)	{3,6}24	0,12	144 / 288 / 432	1,1	ξ'	R1.t12-12	(see ser ξ')
{3,6}(0,14)	{3,6}42	7,7	147 / 294 / 441	1,1	ο'	R1.t0-14	(see ser ο')
{3,6}(13,13)	{3,6}26	0,13	169 / 338 / 507	1,1	ξ'	R1.t13-13	(see ser ξ')
{3,6}(0,16)	{3,6}48	8,8	192 / 384 / 576	1,1	ο'	R1.t0-16	(see ser ο')
{3,6}(14,14)	{3,6}28	0,14	196 / 392 / 588	1,1	ξ'	R1.t14-14	(see ser ξ')
{3,6}(15,15)	{3,6}30	0,15	225 / 450 / 675	1,1	ξ'	R1.t15-15	(see ser ξ')
{3,6}(0,18)	{3,6}	9,9	243 / 486 / 739	1,1	ο'	R1.t0-18	(see ser ο')
{3,6}(16,16)	{3,6}32	0,16	256 / 512 / 768	1,1	ξ'	R1.t16-16	(see ser ξ')
{3,6}(17,17)	{3,6}34	0,17	289 / 578 / 867	1,1	ξ'	R1.t17-17	(see ser ξ')
{3,6}(0,20)	{3,6}60	10,10	300 / 600 / 900	1,1	ο'	R1.t0-20	(see ser ο')
{3,6}(18-18)	{3,6}36	0,0	324 / 648 / 972	1,1	ξ'	R1.t18-18	(see ser ξ')

A {3,6} with Eisenstein values i, j has Petrie polygons with 2(ii+ij+jj) / gcd(i, j) edges.

There are separate pages for other regular maps of genus 1 showing squares only hexagons only.

Other regular maps in the torus

Other Regular Maps

General Index