Yeah, I know about diagonalizing, but maybe I don't know enough about other eigenstuff. Anyhow, I think the issue is with Mathematica handling stuff, I'll try it again when I get version 6.

So, through dinner, I set my computer to compute the explicit, exact eigenvalues of the transition matrix. The results were so surprising, I actually laughed out loud at their beauty:

Of the 170 eigenvalues, exactly90are non-zero. That is certainly not a coincidence; there are 90 shapes, and hence 90 unique rows. (From minors and determinants, I can imagine that you get just the right amount of zeros. but it's still pretty, and some time I'd like to fully understand why.)

Anyhow, the 90 non-zero eigenvalues are of three forms:

- 1 (The 1 root of 1-x)
- The 64 roots of - 4991442219 + 386948115252*x + 138762924658242*x^2 - 3103222624916364*x^3 - 877640411309612746*x^4 + 6767666047147902260*x^5 + 2012337578608379698422*x^6 - 8958764174373753804464*x^7 - 2105450969798079776924812*x^8 + 11046600222902203991043312*x^9 + 1123406093781088176157756896*x^10 - 8114779285022492769604795424*x^11 - 325472049644439264188809738432*x^12 + 2962546562996861776481954741760*x^13 + 56287388574936353280108941504768*x^14 - 629949208560303852773245517141504*x^15 - 6105099366584694343157749747552256*x^16 + 86874786230163907381580006692296704*x^17 + 406640998126424132369607623831846912*x^18 - 8273097635823613171991383746279940096*x^19 - 12847215092645959516368232004260052992*x^20 + 563542116540418623958067303673961512960*x^21 - 380249727800812172360337111385392087040*x^22 - 27876394040223235000410822110796320866304*x^23 + 70200985683043152366609324564260935958528*x^24 + 995061174735599595418926716008043747737600*x^25 - 4377190735910344574289281403129530741161984*x^26 - 24483749753436884219275245277429039028502528*x^27 + 172693480479595766346198778149152449118601216*x^28 + 344268199358502700321288117865500491022073856*x^29 - 4765625577836958680705390357889539456775487488*x^3 0 + 772858710665023800732181764002139803146518528*x^31 + 93400462329952268786337346115591965649619910656*x^ 32 - 173918077700241084337687739506139579202876211200*x ^33 - 1244927272170148943873781598261313236445492674560* x^34 + 4715876999443357796546772004551277481760550551552* x^35 + 9204457647103604719171194562707870353618733367296* x^36 - 74472880131178952274776792185662826546928673619968 *x^37 + 17443350194669512509483766280210174159010360459264 *x^38 + 75191391004923599087061417545533284299211717948211 2*x^39 - 13927791249227776078240490903762146831911538246287 36*x^40 - 43280920416000808833869800194972791652726169833308 16*x^41 + 18110807337525247976881146547787829180504891067465 728*x^42 + 26446806236562635365428496318374472606945724250193 92*x^43 - 12495775864656449852395485032743618177001077355210 3424*x^44 + 18148554882138983236427680894317657290624795908295 8848*x^45 + 40633112472130591883866628308152868033344685251074 4576*x^46 - 15481266283450907844717709600230446897185206707756 72832*x^47 + 54971332503775711093793716428432978484848488873145 1392*x^48 + 55522495833400005878653420117344785319027260232082 39104*x^49 - 10580714260899344013132135083176015760588301879893 557248*x^50 - 25105401045309405161267504284860436825811213099791 81056*x^51 + 35729814707763782643150469325553615343782749037989 462016*x^52 - 47192725438731643771622400943584186952215159805015 228416*x^53 - 14671327554181472737539318334576376054183452695928 504320*x^54 + 12167762673059436924667104845242709903144850159253 1140608*x^55 - 15352842715520548924920201552945956831797069825261 4369280*x^56 + 31072468697838310383691091990364643611438025213753 688064*x^57 + 16618470174861176831016701006962041571324603760502 8462592*x^58 - 27899649455767723350693763788559080462889847255743 2274944*x^59 + 24487369234780003959879339853438500289437687257896 0588800*x^60 - 13677849536791490721479177714902447832424297082721 0883072*x^61 + 49139518222010279228071785788096857437263861359797 338112*x^62 - 10462695095295681452359646651435512209985692421953 945600*x^63 + 10111597944446833081475094750380629249919058448062 87360*x^64
- The 25 roots of -41864 + 3136179*x + 386613206*x^2 - 16868236200*x^3 - 402432346544*x^4 + 21212854848768*x^5 - 58429160850880*x^6 - 5476338956938432*x^7 + 47709591633416832*x^8 + 458983451386437120*x^9 - 6981888874709569536*x^10 - 1163160407514488832*x^11 + 392768286252547276800*x^12 - 1520451906669906886656*x^13 - 7565285380895954436096*x^14 + 68553662577750099099648*x^15 - 78233310032907038883840*x^16 - 913347325546567116521472*x^17 + 4118199373346723755720704*x^18 - 3435110820984016104062976*x^19 - 24441765912940143332818944*x^20 + 99523299112333703179665408*x^21 - 183592589424988418224422912*x^22 + 191248395143239778961457152*x^23 - 108940123945387453071753216*x^24 + 26498949067796948044480512*x^25

Okay, someone explain that.

(Note: This makes 90 roots: 1^2+5^2+8^2. Why squares?)

There is a lot of fascinating math about Square-1.

There's those equations, and I have no idea about the coefficients yet.

And I still don't know why the 613 applies to the eventual distribution. Maybe there's something to do with shape symmetries?

## Bookmarks