Regular hexagonal tilings of the torus

Regular hexagonal tilings of the torus have traditional subscripted labels with the format "(a,b)", with a≤b and a+b even. The number of hexagons in a tiling is (a²+3b²)/4.

A cleaner way of labelling the tilings uses labels with the format "(i,j)", with their order irrelevant. The number of hexagons 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
{6,3}(1,1)	{6,3}2	0,1	2 / 1 / 3	3,6	α' δ ξ	R1.t1-1′	5
{6,3}(0,2)	{6,3}6	1,1	6 / 3 / 9	1,3	ο ο°	R1.t0-2′	1
{6,3}(2,2)	{6,3}4	0,2	8 / 4 / 12	1,2	ξ	R1.t2-2′	1
the Heawood map	{6,3}14	1,2	14 / 7 / 21	1,1		C1.t1-3′	1
{6,3}(3,3)	{6,3}6	0,3	18 / 9 / 27	1,1	ξ ξ°	R1.t3-3′	1
{6,3}(0,4)	{6,3}12	2,2	24 / 12 / 36	1,1	ο	R1.t0-4′	1
{6,3}(2,4)	{6,3}26	1,3	26 / 13 / 39	1,1		C1.t2-4′	1
{6,3}(4,4)	{6,3}8	0,4	32 / 16 / 48	1,1	ξ	R1.t4-4′	1
{6,3}(1,5)	{6,3}38	2,3	38 / 19 / 57	1,1		C1.t1-5′	1
{6,3}(3,5)	{6,3}42	1,4	42 / 21 / 63	1,1		C1.t3-5′	1
{6,3}(5,5)	{6,3}10	0,5	50 / 25 / 75	1,1	ξ	R1.t5-5′	1
{6,3}(0,6)	{6,3}18	3,3	54 / 27 / 81	1,1	ο	R1.t0-6′	1
{6,3}(2,6)	{6,3}28	2,4	56 / 28 / 84	1,1		C1.t2-6′	1
{6,3}(4,6)	{6,3}62	1,5	62 / 31 / 93	1,1		C1.t4-6′	1
{6,3}(6,6)	{6,3}12	0,6	72 / 36 / 108	1,1	ξ	R1.t6-6′	1
{6,3}(1,7)	{6,3}74	3,4	74 / 37 / 111	1,1		C1.t1-7′	1
{6,3}(3,7)	{6,3}78	2,5	78 / 39 / 117	1,1		C1.t3-7′	1
{6,3}(5,7)	{6,3}86	1,6	86 / 43 / 129	1,1		C1.t5-7′	1
{6,3}(0,8)	{6,3}24	4,4	96 / 48 / 144	1,1	ο	R1.t0-8′	1
{6,3}(7,7)	{6,3}14	0,7	98 / 49 / 147	1,1	ξ	R1.t7-7′	1
{6,3}(2,8)	{6,3}98	3,5	98 / 49 / 147	1,1		C1.t2-8′	1
{6,3}(8,8)	{6,3}16	0,8	128 / 64 / 192	1,1	ξ	R1.t8-8′	0
{6,3}(0,10)	{6,3}30	5,5	150 / 75 / 225	1,1	ο	R1.t0-10′	0
{6,3}(9,9)	{6,3}18	0,9	162 / 81 / 243	1,1	ξ	R1.t9-9′	0
{6,3}(10,10)	{6,3}20	0,10	200 / 100 / 300	1,1	ξ	R1.t10-10′	0
{6,3}(0,12)	{6,3}36	6,6	216 / 108 / 324	1,1	ο	R1.t0-12′	0
{6,3}(11,11)	{6,3}22	0,11	242 / 121 / 363	1,1	ξ	R1.t11-11′	0
{6,3}(12,12)	{6,3}24	0,12	288 / 144 / 432	1,1	ξ	R1.t12-12′	0
{6,3}(0,14)	{6,3}42	7,7	294 / 147 / 441	1,1	ο	R1.t0-14′	0
{6,3}(13,13)	{6,3}26	0,13	338 / 169 / 507	1,1	ξ	R1.t13-13′	0
{6,3}(0,16)	{6,3}48	8,8	384 / 192 / 576	1,1	ο	R1.t0-16′	0
{6,3}(14,14)	{6,3}28	0,14	392 / 196 / 588	1,1	ξ	R1.t14-14′	0
{6,3}(15,15)	{6,3}30	0,15	450 / 225 / 675	1,1	ξ	R1.t15-15′	0
{6,3}(0,18)	{6,3}54	9,9	486 / 243 / 739	1,1	ο	R1.t0-18′	0
{6,3}(16,16)	{6,3}32	0,16	512 / 256 / 768	1,1	ξ	R1.t16-16′	0
{6,3}(17,17)	{6,3}34	0,17	578 / 289 / 867	1,1	ξ	R1.t17-17′	0
{6,3}(0,20)	{6,3}	10,10	600 / 300 / 900	1,1	ο	R1.t0-20′	0
{6,3}(18,18)	{6,3}36	0,18	648 / 324 / 972	1,1	ξ	R1.t18-18′	0

A {6,3} 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 and triangles only.

Other regular maps in the torus

Other Regular Maps

General Index