### Aperiodic Hat Transformations ðŸŽ©

Am I doing this right?

In Aperiodic Hat Monotiles the 'Hat' tile is used in a tessellation. Using different ratios for the length of the edges an infinite amount of different shapes are possible for this tessellation (youtu.be/w-ecvtia-5a).

Use the slide for t_morph to 'scroll' through these possibilities. It is a scalar from 0 to 6 (instead of 0 to 1) because there are 6 intervals between the 7 predefined criteria (see lines 56 to 67). The 'hat' is on the 1/3 interval which cannot be expressed with a slider from 0 to 1. With values from 0 to 6 the 'hat' is at value 2.

Using the shoelace approach I solved the problem to keep the area's of the different tile variations the same (lines 86 to 94). However I yet failed to integrate these variations in Aperiodic Hat Monotiles (see Fork: Aperiodic Hat Monotiles) because some transformations are applied to polygons that I do not yet fully understand.

cs.uwaterloo.ca/~csk/hat/
arxiv.org/pdf/2303.10798.pdf

Log in to post a comment.

```const t_morph = 2; //min=0 max=6 step=.001
const scale = 20; //--min=0 max=80 step=.5

// You can find the Turtle API reference here: https://turtletoy.net/syntax
Canvas.setpenopacity(1);

// Global code will be evaluated once.
const turtle = new Turtle();
const polygons = new Polygons();
const text = new Text();

const r3 = Math.sqrt(3);
const hr3 = r3 / 2;

const scl2   = (a,b)   => [a[0]*b, a[1]*b];
const add2   = (a,b)   => [a[0]+b[0], a[1]+b[1]];
const sub2   = (a,b)   => [a[0]-b[0], a[1]-b[1]];
const rot2   = (a)     => [Math.cos(a), -Math.sin(a), Math.sin(a), Math.cos(a)];
const hexPt  = (x, y)  => [x + 0.5 * y, hr3 * y];

function walk(i) {
const hat_outline = morph_tile_outline(t_morph);
const centerCorrection = scl2(getPolygonCenter(hat_outline), -1);

const p = polygons.create();
p.addPoints(...hat_outline.map(pt => scl2(add2(centerCorrection, pt), scale)));
p.addOutline();
polygons.draw(turtle, p);

displayShapeName(t_morph);

return false;
}

function displayShapeName(t_morph) {
const namingPrecision = 5;
if(Math.round(namingPrecision * t_morph /2) / namingPrecision % 1 == 0) {
const names = ['Chevron', 'Hat', 'Turtle', 'Comet'];
let width = 0;
for(let i = -1; i < 1; i++) {
turtle.jump(0 - width, -85 - i * 300);
text.print(turtle, '- ' + names[Math.round(t_morph /2)] + ' -', .5);
width = turtle.pos()[0] / 2;
}
}
};

function morph_tile_outline(morph_t = 2) {
const hat_outline = [
hexPt(0, 0), hexPt(-1, -1), hexPt(0, -2), hexPt(2, -2),
hexPt(2, -1), hexPt(4, -2), hexPt(5, -1), hexPt(4, 0),
hexPt(3, 0), hexPt(2, 2), hexPt(0, 3), hexPt(0, 2),
hexPt(-1, 2)
];

// criteria taken from figure 2.3 in https://arxiv.org/pdf/2303.10798.pdf
//  and it's also mentioned in https://isohedral.ca/aperiodic-monotiles/
//  as https://isohedral.ca/wp-content/uploads/2023/12/tile_ab-1024x188.png
const criteria = [
[0, 1],      //t_morph = 0  Chevron
[1, 4],      //t_morph = 1
[1, 3**.5],  //t_morph = 2  Hat
[1, 1],      //t_morph = 3
[3**.5, 1],  //t_morph = 4  Turtle
[4, 1],      //t_morph = 5
[1, 0],      //t_morph = 6  Comet
];

const lerp2  = (a,b,t) => a.map((v, i) => v*(1-t) + b[i]*t);

const startIndex = t_morph== 6? criteria.length - 2: ((t_morph * (criteria.length - 1) / 6)|0);
const endIndex   = t_morph== 6? criteria.length - 1: ((t_morph * (criteria.length - 1) / 6)|0) + 1;

const [a, b] = lerp2(criteria[startIndex], criteria[endIndex], t_morph == 6? 1: (t_morph * (criteria.length - 1) / 6) % 1);
const tile_edgetypes = [b, a, 2 * a, a, b, b, a, a, b, b, a, a, b];

const tile_outline = hat_outline.map((e,i,a) => sub2(a[(i+1)%a.length], e)) //map hat_outline vertices to vectors representing vectors from vertice to next vertice
.map((e,i) => scl2(e, tile_edgetypes[i] / Math.hypot(...e))) //scale each edge to ratio in tile_edgetypes
.reduce((a, c) => [...a, add2(a[a.length-1], c)], [[0,0]]); //and map those scaled edges back to vertices
tile_outline.pop(); // conversion to vectors and back causes last point identical to start point to be added to tile_outline

const areaScalar = getVerticeScalarByArea(getPolgyonArea(tile_outline), getPolgyonArea(hat_outline));
return tile_outline.map(pt => scl2(pt, areaScalar));
}

//Shoelace https://en.wikipedia.org/wiki/Shoelace_formula
function getPolgyonArea(vertices) {
const shoelace = (a, b) => a[0] * b[1] - a[1] * b [0];
return vertices.reduce((a, c, i) => a + shoelace(c, vertices[(i+1)%vertices.length]), 0);
}

function getVerticeScalarByArea(area, normalizeToArea = 1) {
return (normalizeToArea / Math.abs(area))**.5;
}

function getPolygonCenter(polygon) {
const minmax = polygon.reduce((a, c) => [[Math.min(c[0], a[0][0]), Math.max(c[0], a[0][1])], [Math.min(c[1], a[1][0]), Math.max(c[1], a[1][1])]], [[Number.MAX_SAFE_INTEGER,Number.MIN_SAFE_INTEGER], [Number.MAX_SAFE_INTEGER,Number.MIN_SAFE_INTEGER]]);
return [minmax[0][0] + (minmax[0][1] - minmax[0][0])/2, minmax[1][0] + (minmax[1][1] - minmax[1][0])/2];
}

////////////////////////////////////////////////////////////////
// Polygon Clipping utility code - Created by Reinder Nijhoff 2019
// (Polygon binning by Lionel Lemarie 2021)
// https://turtletoy.net/turtle/a5befa1f8d
////////////////////////////////////////////////////////////////
function Polygons(){const t=[],s=25,e=Array.from({length:s**2},t=>[]),n=class{constructor(){this.cp=[],this.dp=[],this.aabb=[]}addPoints(...t){let s=1e5,e=-1e5,n=1e5,h=-1e5;(this.cp=[...this.cp,...t]).forEach(t=>{s=Math.min(s,t[0]),e=Math.max(e,t[0]),n=Math.min(n,t[1]),h=Math.max(h,t[1])}),this.aabb=[s,n,e,h]}addSegments(...t){t.forEach(t=>this.dp.push(t))}addOutline(){for(let t=0,s=this.cp.length;t<s;t++)this.dp.push(this.cp[t],this.cp[(t+1)%s])}draw(t){for(let s=0,e=this.dp.length;s<e;s+=2)t.jump(this.dp[s]),t.goto(this.dp[s+1])}addHatching(t,s){const e=new n;e.cp.push([-1e5,-1e5],[1e5,-1e5],[1e5,1e5],[-1e5,1e5]);const h=Math.sin(t)*s,o=Math.cos(t)*s,a=200*Math.sin(t),i=200*Math.cos(t);for(let t=.5;t<150/s;t++)e.dp.push([h*t+i,o*t-a],[h*t-i,o*t+a]),e.dp.push([-h*t+i,-o*t-a],[-h*t-i,-o*t+a]);e.boolean(this,!1),this.dp=[...this.dp,...e.dp]}inside(t){let s=0;for(let e=0,n=this.cp.length;e<n;e++)this.segment_intersect(t,[.1,-1e3],this.cp[e],this.cp[(e+1)%n])&&s++;return 1&s}boolean(t,s=!0){const e=[];for(let n=0,h=this.dp.length;n<h;n+=2){const h=this.dp[n],o=this.dp[n+1],a=[];for(let s=0,e=t.cp.length;s<e;s++){const n=this.segment_intersect(h,o,t.cp[s],t.cp[(s+1)%e]);!1!==n&&a.push(n)}if(0===a.length)s===!t.inside(h)&&e.push(h,o);else{a.push(h,o);const n=o[0]-h[0],i=o[1]-h[1];a.sort((t,s)=>(t[0]-h[0])*n+(t[1]-h[1])*i-(s[0]-h[0])*n-(s[1]-h[1])*i);for(let n=0;n<a.length-1;n++)(a[n][0]-a[n+1][0])**2+(a[n][1]-a[n+1][1])**2>=.001&&s===!t.inside([(a[n][0]+a[n+1][0])/2,(a[n][1]+a[n+1][1])/2])&&e.push(a[n],a[n+1])}}return(this.dp=e).length>0}segment_intersect(t,s,e,n){const h=(n[1]-e[1])*(s[0]-t[0])-(n[0]-e[0])*(s[1]-t[1]);if(0===h)return!1;const o=((n[0]-e[0])*(t[1]-e[1])-(n[1]-e[1])*(t[0]-e[0]))/h,a=((s[0]-t[0])*(t[1]-e[1])-(s[1]-t[1])*(t[0]-e[0]))/h;return o>=0&&o<=1&&a>=0&&a<=1&&[t[0]+o*(s[0]-t[0]),t[1]+o*(s[1]-t[1])]}};return{list:()=>t,create:()=>new n,draw:(n,h,o=!0)=>{reducedPolygonList=function(n){const h={},o=200/s;for(var a=0;a<s;a++){const c=a*o-100,r=[0,c,200,c+o];if(!(n[3]<r[1]||n[1]>r[3]))for(var i=0;i<s;i++){const c=i*o-100;r[0]=c,r[2]=c+o,n[0]>r[2]||n[2]<r[0]||e[i+a*s].forEach(s=>{const e=t[s];n[3]<e.aabb[1]||n[1]>e.aabb[3]||n[0]>e.aabb[2]||n[2]<e.aabb[0]||(h[s]=1)})}}return Array.from(Object.keys(h),s=>t[s])}(h.aabb);for(let t=0;t<reducedPolygonList.length&&h.boolean(reducedPolygonList[t]);t++);h.draw(n),o&&function(n){t.push(n);const h=t.length-1,o=200/s;e.forEach((t,e)=>{const a=e%s*o-100,i=(e/s|0)*o-100,c=[a,i,a+o,i+o];c[3]<n.aabb[1]||c[1]>n.aabb[3]||c[0]>n.aabb[2]||c[2]<n.aabb[0]||t.push(h)})}(h)}}}

////////////////////////////////////////////////////////////////
// Text utility code. Created by Reinder Nijhoff 2019
// https://turtletoy.net/turtle/1713ddbe99
// Jurgen 2021: Fixed Text.print() to restore turtle._fullCircle
//.             if was in e.g. degrees mode (or any other)
////////////////////////////////////////////////////////////////
function Text() {class Text {print (t, str, scale = 1, italic = 0, kerning = 1) {let fc = t._fullCircle;t.radians();let pos = [t.x(), t.y()], h = t.h(), o = pos;str.split('').map(c => {const i = c.charCodeAt(0) - 32;if (i < 0 ) {pos = o = this.rotAdd([0, 48*scale], o, h);} else if (i > 96 ) {pos = this.rotAdd([16*scale, 0], o, h);} else {const d = dat[i], lt = d[0]*scale, rt = d[1]*scale, paths = d[2];paths.map( p => {t.up();p.map( s=> {t.goto(this.rotAdd([(s[0]-s[1]*italic)*scale - lt, s[1]*scale], pos, h));t.down();});});pos = this.rotAdd([(rt - lt)*kerning, 0], pos, h);}});t._fullCircle = fc;}rotAdd (a, b, h) {return [Math.cos(h)*a[0] - Math.sin(h)*a[1] + b[0], Math.cos(h)*a[1] + Math.sin(h)*a[0] + b[1]];}}const dat = ('br>eoj^jl<jqirjskrjq>brf^fe<n^ne>`ukZdz<qZjz<dgrg<cmqm>`thZhw<lZlw<qao_l^h^e_caccdeefggmiojpkqmqporlshsercp>^vs^as<f^h`hbgdeeceacaab_d^f^h_k`n`q_s^<olmmlolqnspsrrspsnqlol>]wtgtfsereqfphnmlpjrhsdsbraq`o`makbjifjekckaj_h^f_eaecffhimporqssstrtq>eoj`i_j^k_kajcid>cqnZl\\j_hcghglhqjulxnz>cqfZh\\j_lcmhmllqjuhxfz>brjdjp<egom<ogem>]wjajs<ajsj>fnkojpiojnkokqis>]wajsj>fnjniojpkojn>_usZaz>`ti^f_dbcgcjdofrisksnrpoqjqgpbn_k^i^>`tfbhak^ks>`tdcdbe`f_h^l^n_o`pbpdofmicsqs>`te^p^jfmfogphqkqmppnrkshserdqco>`tm^clrl<m^ms>`to^e^dgefhekenfphqkqmppnrkshserdqco>`tpao_l^j^g_ebdgdlepgrjsksnrppqmqlpingkfjfggeidl>`tq^gs<c^q^>`th^e_dadceegfkgnhpjqlqopqorlshserdqcocldjfhigmfoepcpao_l^h^>`tpeohmjjkikfjdhcecddaf_i^j^m_oapepjoomrjshserdp>fnjgihjikhjg<jniojpkojn>fnjgihjikhjg<kojpiojnkokqis>^vrabjrs>]wagsg<amsm>^vbarjbs>asdcdbe`f_h^l^n_o`pbpdofngjijl<jqirjskrjq>]xofndlcicgdfeehekfmhnknmmnk<icgefhfkgmhn<ocnknmpnrntluiugtdsbq`o_l^i^f_d`bbad`g`jambodqfrislsorqqrp<pcokompn>asj^bs<j^rs<elol>_tc^cs<c^l^o_p`qbqdpfoglh<chlhoipjqlqopqorlscs>`urcqao_m^i^g_eadccfckdnepgrismsorqprn>_tc^cs<c^j^m_oapcqfqkpnopmrjscs>`sd^ds<d^q^<dhlh<dsqs>`rd^ds<d^q^<dhlh>`urcqao_m^i^g_eadccfckdnepgrismsorqprnrk<mkrk>_uc^cs<q^qs<chqh>fnj^js>brn^nnmqlrjshsfreqdndl>_tc^cs<q^cl<hgqs>`qd^ds<dsps>^vb^bs<b^js<r^js<r^rs>_uc^cs<c^qs<q^qs>_uh^f_daccbfbkcndpfrhslsnrppqnrkrfqcpan_l^h^>_tc^cs<c^l^o_p`qbqepgohlici>_uh^f_daccbfbkcndpfrhslsnrppqnrkrfqcpan_l^h^<koqu>_tc^cs<c^l^o_p`qbqdpfoglhch<jhqs>`tqao_l^h^e_caccdeefggmiojpkqmqporlshsercp>brj^js<c^q^>_uc^cmdpfrisksnrppqmq^>asb^js<r^js>^v`^es<j^es<j^os<t^os>`tc^qs<q^cs>asb^jhjs<r^jh>`tq^cs<c^q^<csqs>cqgZgz<hZhz<gZnZ<gznz>cqc^qv>cqlZlz<mZmz<fZmZ<fzmz>brj\\bj<j\\rj>asazsz>fnkcieigjhkgjfig>atpeps<phnfleiegfehdkdmepgrislsnrpp>`sd^ds<dhffhekemfohpkpmopmrkshsfrdp>asphnfleiegfehdkdmepgrislsnrpp>atp^ps<phnfleiegfehdkdmepgrislsnrpp>asdkpkpiognfleiegfehdkdmepgrislsnrpp>eqo^m^k_jbjs<gene>atpepuoxnylzizgy<phnfleiegfehdkdmepgrislsnrpp>ate^es<eihfjemeofpips>fni^j_k^j]i^<jejs>eoj^k_l^k]j^<kekvjyhzfz>are^es<oeeo<ikps>fnj^js>[y_e_s<_ibfdegeifjijs<jimfoeretfuius>ateees<eihfjemeofpips>atiegfehdkdmepgrislsnrppqmqkphnfleie>`sdedz<dhffhekemfohpkpmopmrkshsfrdp>atpepz<phnfleiegfehdkdmepgrislsnrpp>cpgegs<gkhhjfleoe>bsphofleieffehfjhkmlompopporlsisfrep>eqj^jokrmsos<gene>ateeeofrhsksmrpo<peps>brdejs<pejs>_ubefs<jefs<jens<rens>bseeps<pees>brdejs<pejshwfydzcz>bspees<eepe<esps>cqlZj[i\\h^h`ibjckekgii<j[i]i_jakbldlfkhgjkllnlpkrjsiuiwjy<ikkmkojqirhthvixjylz>fnjZjz>cqhZj[k\\l^l`kbjcieigki<j[k]k_jaibhdhfihmjilhnhpirjskukwjy<kkimiojqkrltlvkxjyhz>^vamakbhdgfghhlknlplrksi<akbidhfhhillnmpmrlsisg>brb^bscsc^d^dsese^f^fsgsg^h^hsisi^j^jsksk^l^lsmsm^n^nsoso^p^psqsq^r^rs').split('>').map(r=> { return [r.charCodeAt(0)-106,r.charCodeAt(1)-106, r.substr(2).split('<').map(a => {const ret = []; for (let i=0; i<a.length; i+=2) {ret.push(a.substr(i, 2).split('').map(b => b.charCodeAt(0)-106));} return ret; })]; });return new Text();}
```