Some results regarding turmites which move on a grid of hexagons.

There are 6 ways to turn, compared to 4 for a square-grid turmite. Therefore, for given numbers of states and colours, there are more hex turmites than square ones. Some of those hex turmites surpass records set by the square turmites.

Ed Pegg's notation for a turmite's rule-set is suitable only if the number representing a turn is a single digit. This is not true for two of the turns for hex turmites, so I invented my own notation. This begins with s for "square" or (as here) h for "hex". I list the elements of a rule-set in the same order as in Golly's format. I write ';' wherever Golly's notation has "}},{{"; thus, if there are s states and c colours, then there are s-1 semicolons separating s substrings of 6c characters each. I label the colours of cells A, B etc. (the better to distinguish them from the values of the turmite's state). I symbolise turns as follows:

My letter	Golly's number	Turn
f	1	forwards
l	2	60° left
r	4	60° right
p	8	120° left
s	16	120° right
u	32	U-turn

For p and s I followed the notation of J Corbin in [hexant](https://github.com/jcorbin/hexant). Admittedly J Corbin uses B for backwards for a U-turn, but I had already chosen u when notating square-grid turmites' rule-sets.

When giving coordinates, I define the positive y-axis as extending in a direction 120 degrees anticlockwise from the positive x-axis. If you define the positive y-axis as extending in a direction 60 degrees anticlockwise from the positive x-axis, then my (x, y) is your (x-y, y).

I start each turmite pointing in the positive x-direction (1, 0), aka "east".

Excluding duplicates, turmites that resolve to periodic behaviour by tick 63, and two turmites each of which builds a 2-way highway right from tick 0, there are 7 non-trivial 1-state 2-colour hex turmites. Two are periodic, with saltuses (x, y) as given below:

My notation	Golly notation	preperiod	period	x	y
hBr0Af0	{{{1,4,0},{0,1,0}}}	4806191	288	-7	-6
hBs0Al0	{{{1,16,0},{0,2,0}}}	5528572	62976	-2	-94

hBr0Al0 {{{1,4,0},{0,2,0}}} builds a shape roughly bounded by a dodecagon with five acute points. I call this shape, and other similar shapes made by similar hex turmites, crowns. Though the outer boundary remains relatively simple, the colouring of the interior always contains chaotic elements. It is notable that, despite all this complexity, the turmite maintains bilateral symmetry. This phenomenon also occurs with some square-grid turmites, but only with turmites with at least four rules, e.g. the 1-state 4-colour turmites sBr0Cr0Dl0Al0 {{{1,2,0},{2,2,0},{3,8,0},{0,8,0}}} and sBr0Cl0Dl0Ar0 {{{1,2,0},{2,8,0},{3,8,0},{0,2,0}}}. This hex turmite achieves this with two rules (1 state and 2 colours).

The other four 1-state 2-colour hex turmites, listed below, are so far unresolved.

My notation	Golly notation	ticks run	notes
hBr0As0	{{{1,4,0},{0,16,0}}}	100G
hBr0Ap0	{{{1,4,0},{0,8,0}}}	100G
hBs0Ar0	{{{1,16,0},{0,4,0}}}	100G
hBs0Bf0	{{{1,16,0},{1,1,0}}}	100G

For a list of 1-state 3-colour hex turmites which I have not resolved, see UnresolvedHexTurmites1s3c.

Among the 1-state 3-colour hex turmites whose behaviour eventually becomes periodic, the following are notable:

My notation	Golly notation	preperiod	period	x	y	notes
hBr0Cs0Al0	{{{1,4,0},{2,16,0},{0,2,0}}}	774591929	100	1	3	Highest preperiod found so far
hBs0Cl0Bf0	{{{1,16,0},{2,2,0},{1,1,0}}}	158742520	119	-3	1
hBr0Cl0Au0	{{{1,4,0},{2,2,0},{0,32,0}}}	0	31452	0	0	After 31452 ticks, the turmite restores its direction and all 24 visited hexes to their starting state, and returns to its origin.
hBu0Cr0Cl0	{{{1,32,0},{2,4,0},{2,2,0}}}	44897	6	0	0	Stuck in a loop of 6 hexes coloured C.

Some more miscellaneous interesting resolved turmites.

hBu0Cr0Bl0 {{{1,32,0},{2,4,0},{1,2,0}}} makes a pattern that is always basically symmetrical about one axis; this becomes gradually more like a Star of David.

hBs0Cs0Cf0 {{{1,16,0},{2,16,0},{2,1,0}}} A 6-pointed star, all in C except that some boundary hexes are B. Macroscopically it has the symmetry of the equilateral triangle, though at a cell level its only symmetry is about the true vertical axis. The boundary of each pair of legs is a discrete approximation to an ellipse, the length of each leg being proportional to the square of the body's width. Its area varies as t^(3/4). The legs' lengths vary as t^(1/2).

For a list of 1-state 4-colour hex turmites which I have not resolved, see UnresolvedHexTurmites1s4c.

Among the 1-state 4-colour hex turmites whose behaviour eventually becomes periodic, the following are notable in taking a long time (>100M) to resolve:

My notation	Golly notation	preperiod	period	x	y	notes
hBr0Cf0Dl0Ar0	{{{1,4,0},{2,1,0},{3,2,0},{0,4,0}}}	1173099793	40	-2	-1
hBr0Cf0Dl0Ap0	{{{1,4,0},{2,1,0},{3,2,0},{0,8,0}}}	5550308320	108	1	4
hBr0Cf0Ds0Af0	{{{1,4,0},{2,1,0},{3,16,0},{0,1,0}}}	5479715065	132	1	4
hBr0Cr0Dr0Af0	{{{1,4,0},{2,4,0},{3,4,0},{0,1,0}}}	1020964081	98	-2	-1
hBr0Cr0Df0Cs0	{{{1,4,0},{2,4,0},{3,1,0},{2,16,0}}}	3965847188	716	-9	0
hBr0Cr0Ds0As0	{{{1,4,0},{2,4,0},{3,16,0},{0,16,0}}}	3561651478	194	-2	0
hBr0Cp0Dr0Af0	{{{1,4,0},{2,8,0},{3,4,0},{0,1,0}}}	842799569	185	5	0
hBr0Cp0Ds0Al0	{{{1,4,0},{2,8,0},{3,16,0},{0,2,0}}}	467433144	137	-4	-2
hBr0Cs0Df0Bu0	{{{1,4,0},{2,16,0},{3,1,0},{1,32,0}}}	2950286036	123	1	-2
hBr0Cs0Dr0Af0	{{{1,4,0},{2,16,0},{3,4,0},{0,1,0}}}	323335518	27	1	1
hBs0Cf0Df0Bp0	{{{1,16,0},{2,1,0},{3,1,0},{1,8,0}}}	425579546	73	-2	0
hBs0Cf0Dp0Cr0	{{{1,16,0},{2,1,0},{3,8,0},{2,4,0}}}	3616327572	70	2	2
hBs0Cl0Df0Af0	{{{1,16,0},{2,2,0},{3,1,0},{0,1,0}}}	4469645622	30	1	1
hBs0Cl0Ds0Bf0	{{{1,16,0},{2,2,0},{3,16,0},{1,1,0}}}	764972819	92	-3	-2
hBs0Cr0Dl0Af0	{{{1,16,0},{2,4,0},{3,2,0},{0,1,0}}}	3112357427	29	0	1
hBs0Cr0Df0Bp0	{{{1,16,0},{2,4,0},{3,1,0},{1,8,0}}}	1623221015	27	0	1
hBs0Cr0Dr0Ar0	{{{1,16,0},{2,4,0},{3,4,0},{0,4,0}}}	705361184	85	-1	-1
hBs0Cr0Ds0Al0	{{{1,16,0},{2,4,0},{3,16,0},{0,2,0}}}	958740998	106	-2	-2
hBs0Cr0Dp0Bu0	{{{1,16,0},{2,4,0},{3,8,0},{1,32,0}}}	5410020710	115	2	0
hBs0Cp0Dl0As0	{{{1,16,0},{2,8,0},{3,2,0},{0,16,0}}}	580868961	50	-1	0
hBs0Cs0Ds0Cr0	{{{1,16,0},{2,16,0},{3,16,0},{2,4,0}}}	103767767	1325	-3	3
hBs0Cu0Dp0Al0	{{{1,16,0},{2,32,0},{3,8,0},{0,2,0}}}	3398213240	632	8	6
hBu0Cs0Dl0Bp0	{{{1,32,0},{2,16,0},{3,2,0},{1,8,0}}}	414163771	49	1	0
hBu0Cs0Dp0As0	{{{1,32,0},{2,16,0},{3,8,0},{0,16,0}}}	391373751	22368	-31	-10
hBu0Cs0Ds0Cp0	{{{1,32,0},{2,16,0},{3,16,0},{2,8,0}}}	1621271084	995	3	2
hBu0Cu0Dr0Ap0	{{{1,32,0},{2,32,0},{3,4,0},{0,8,0}}}	552210023	1248	-7	-5

Among these periodic turmites, the one of greatest period is hBs0Cp0Du0Al0 {{{1,16,0},{2,8,0},{3,32,0},{0,2,0}}}, of preperiod 45047511 and period 352688. Its saltus of -8,133 (length ~137) is also the longest.

The turmite hBu0Cr0Du0Bl0 {{{1,32,0},{2,4,0},{3,32,0},{1,2,0}}}, at tick 9075991, for the 96th and last time, visits a new hex. From this point on, it is trapped: its behaviour has period 139572 and saltus 0,0.

Several of these turmites make double highways. Of these, hBr0Cs0Ds0Df0 {{{1,4,0},{2,16,0},{3,16,0},{3,1,0}}} is notable in not resolving to double-highway-building until ~9,475M ticks, a record which surpasses all the above turmites which make classic single highways.

hBr0Cl0Dl0Bu0 {{{1,4,0},{2,2,0},{3,2,0},{1,32,0}}} makes a pattern within a basically regular hexagonal boundary, mostly B and C. Most B/C boundaries are parallel to the nearest side of the outer boundary, but there are occasional chaotic regions.

hBr0Cr0Dp0Ap0 {{{1,4,0},{2,4,0},{3,8,0},{0,8,0}}} makes a pattern, of density 3/4, macroscopically symmetrical about an axis in a direction at right angles to the turmite's initial direction, just like sBr0Cr0Dl0Al0 and sBr0Cl0Dl0Ar0. It grows within a slowly-growing hexagonal boundary.

hBs0Cr0Du0Df0 {{{1,16,0},{2,4,0},{3,32,0},{3,1,0}}} makes two diametrically-opposed 60-degree wedges in D by ~4,160,000 ticks.

hBs0Cs0Df0Cf0 {{{1,16,0},{2,16,0},{3,1,0},{2,1,0}}} builds the same 6-legged shape as hBs0Cs0Cf0 {{{1,16,0},{2,16,0},{2,1,0}}}, but with the hexes coloured C and D in an interesting pattern.

hBs0Cs0Dl0Al0 {{{1,16,0},{2,16,0},{3,2,0},{0,2,0}}} makes a pattern, of density 3/4, macroscopically symmetrical about an axis in a direction at right angles to the turmite's initial direction. Its pattern's asymptotic shape, to judge from the shape at 10G, appears to be similar to that of sBr0Cr0Dl0Al0 t14r120220380080 {{{1,2,0},{2,2,0},{3,8,0},{0,8,0}}}.

It seems that each of the following turmites eventually ceases to visit new hexes, but continues to behave chaotically thereafter; until tick 10G, at any rate. This cannot be actually true, of course; the turmite must either continually visit new hexes, or else get trapped.

hBr0Cr0Dl0Bu0 {{{1,4,0},{2,4,0},{3,2,0},{1,32,0}}} By tick 39830964 it has visited 9328 hexes. Indeed, after its return to its origin at tick 39840360, it rarely ventures far from there.

hBr0Cu0Df0Bf0 {{{1,4,0},{2,32,0},{3,1,0},{1,1,0}}} This turmite builds a chaotic interior, and also frequently grows hexagonal spirals rooted to spots on the boundary. The building of each such spiral typically stops when it runs into something else. One particular spiral grows quite big by 1G. By 3G it has visited 140123 hexes. By 100G it has not visited any more -- is it now trapped?

hBr0Cu0Du0Al0 {{{1,4,0},{2,32,0},{3,32,0},{0,2,0}}} By tick 20177, it has visited 72 hexes.

Each 2-state 2-colour hex turmite has been run until resolved or for 1G ticks. Of these turmites, over 7000 are not resolved by 1G ticks.

Among the 2-state 2-colour hex turmites whose behaviour eventually becomes periodic, the following are notable.

(If the unresolved turmites were run further, it is very probable that some of them would beat these records.)

My notation	Golly notation	preperiod	period	x	y	notes
hAs1As0;Bp1Ar0	{{{0,16,1},{0,16,0}},{{1,8,1},{0,4,0}}}	114096255	10401634	720	115	Longest period known.
hAr1Af0;Bl1Ap0	{{{0,4,1},{0,1,0}},{{1,2,1},{0,8,0}}}	230798188	8727997	-245	1204	Longest saltus known: sqrt 1804621 ~ 1343
hBr1As1;Bu0Ar0	{{{1,4,1},{0,16,1}},{{1,32,0},{0,4,0}}}	410428084	1364196	-255	-297
hBf1Ar0;Br0As1	{{{1,1,1},{0,4,0}},{{1,4,0},{0,16,1}}}	986640707	1530	-4	11	Longest preperiod known.

Here are some interesting turmites from among the resolved ones.

My notation	Golly notation	notes
hAr1Af0;Bp0As1	{{{0,4,1},{0,1,0}},{{1,8,0},{0,16,1}}}	90° wedge
hAs1Bu1;Bf0Bp0	{{{0,16,1},{1,32,1}},{{1,1,0},{1,8,0}}}	Four 5-hex-wide highways.
hBr0Bf1;Bf0Au0	{{{1,4,0},{1,1,1}},{{1,1,0},{0,32,0}}}	Highway after several attempts at highways and wedges.
hBr0Bf1;Bu0Af0	{{{1,4,0},{1,1,1}},{{1,32,0},{0,1,0}}}	Two opposing 60° wedges
hBs0Af1;Bp1Bf0	{{{1,16,0},{0,1,1}},{{1,8,1},{1,1,0}}}	60° wedge