### Escher-ish cubes ðŸ“¦

Isometric impossible objects*

Escher-ish cubes ðŸ“¦ (variation)
Escher-ish cubes ðŸ“¦ (variation)
Escher-ish cubes ðŸ“¦ (variation)
Escher-ish cubes ðŸ“¦ (variation)

(*Except with tileRings = 0 which yields a possible object)

```// You can find the Turtle API reference here: https://turtletoy.net/syntax
Canvas.setpenopacity(.75);

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

const sq3 = Math.sqrt(3);
const hsq3 = sq3/2;

const tileRings = 1; //min=0 max=15 step=1
const tileHeight = 11.5; //min=2 max=40 step=.5
const gapRatio = 4; //min=1 max=6 step=.5
const edgeCubes = 4; //min=2 max=10 step=2
const clipStubs = 1; //min=0 max=1 step=1 (No, Yes)
const forPlotter = 0; //min=0 max=1 step=1 (No: add hatching, Yes: but polygons util sometimes get outLines wrong on different gapRatios)

const gap = gapRatio == 6? 0: tileHeight / gapRatio;
const height = tileHeight;
const clipTails = clipStubs == 1;

const scl = (p, s) => [p[0] * s, p[1] * s];
const add = (a, b) => [a[0]+b[0], a[1]+b[1]];
const trns = (a) => [a[0] * ((height + gap) * hsq3) , a[1] * ((height + gap) / 2)];

let blockParams = [];
let cells = [];
// The walk function will be called until it returns false.
function walk(i) {
if(i == 0) {
for(let j = 0; j <= hg.nthTriangular(tileRings) * 6; j++) {
cells.push(hg.nextCell());
}
let topCells = cells.filter((i) => i.cube[2] == tileRings || i.cube[1] == -tileRings).map((i) => i.index);

let noTopleft = cells.filter((i) => i.cube[2] == tileRings || i.cube[0] == -tileRings).map((i) => i.index);
let noTopRight = cells.filter((i) => i.cube[1] == -tileRings || i.cube[0] == tileRings).map((i) => i.index);
let noBottom = cells.filter((i) => i.cube[1] == tileRings || i.cube[2] == -tileRings).map((i) => i.index);
let noBottomRight = cells.filter((i) => i.cube[2] == -tileRings || i.cube[0] == tileRings).map((i) => i.index);
let noBottomLeft = cells.filter((i) => i.cube[1] == tileRings || i.cube[0] == -tileRings).map((i) => i.index);

cells.sort((a,b) => (a.double[1] > b.double[1]) ? -1 : ((b.double[1] > a.double[1]) ? 1 : 0))

cells.forEach(function(cell) {
paramIntersection(cell.position, clipTails&&noBottomLeft.includes(cell.index)? 0: edgeCubes/2, clipTails&&noBottomRight.includes(cell.index)? 0: edgeCubes/2, clipTails&&topCells.includes(cell.index)? 0: edgeCubes/2, clipTails&&noBottom.includes(cell.index)? 0:  edgeCubes/2, clipTails&&noTopRight.includes(cell.index)? 0: edgeCubes/2, clipTails&&noTopleft.includes(cell.index)? 0:edgeCubes/2, !topCells.includes(cell.index));
});

// cells.forEach(function(cell) { turtle.jump(cell.position);cell.drawBorder(turtle); });
return true;
}

let block = blockParams.pop();
drawCube(turtle, ...block);

return blockParams.length > 0;
}

function paramIntersection(trans, downleft, downright, up, down, upright, upleft, cheatUp = false) {
cheatUp = cheatUp? gap/height: false;
for(let u = downleft; u > 0; u--) { //downleft to mid
blockParams.unshift([[-u, u], height, trans]);
}
for(let u = downright; u > 0; u--) { //downright to mid
blockParams.unshift([[u, u], height, trans]);
}
for(let u = -up; u <= down; u++) { //up to down
blockParams.unshift([[0, 2 * u], height, trans, cheatUp !== false && u == -up? cheatUp: false]);
}
for(let u = -1; u >= -upright; u--) { //mid to upright
blockParams.unshift([[-u, u], height, trans]);
}
for(let u = -1; u >= -upleft; u--) { //mid to upleft
blockParams.unshift([[u, u], height, trans]);
}
}

function drawCube(t, pos, size, transpose, cheatTop = false) {
const hsq3 = Math.sqrt(3)/2;
let faces = [
[[[0, 0],[hsq3, -.5],[0, -1],[-hsq3, -.5]],[Math.PI/6, .8]],//top
[[[0, 0],[0 + hsq3, -.5],[0 + hsq3, .5],[0, 1]],[1, .5]],  // right
[[[0, 0],[0 - hsq3, -.5],[0 - hsq3, .5],[0, 1]],[2, .25]]   //left
];

if(cheatTop !== false) {
faces[0][0] = [[0, 0],[hsq3, -.5],[hsq3 * (1-cheatTop), -.5 * (1+cheatTop)],[0, -cheatTop],[-hsq3 * (1-cheatTop), -.5 * (1+cheatTop)],[-hsq3, -.5]];
}

faces.forEach(function(f) {
let p = polygons.create();
if(forPlotter == 0) {
} else {
}
polygons.draw(t, p);
});
}

////////////////////////////////////////////////////////////////
// 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)}}}

////////////////////////////////////////////////////////////////
// Hexagon Grid utility code - Created by Jurgen Westerhof 2021
// https://turtletoy.net/turtle/906427a302
////////////////////////////////////////////////////////////////
class HexGrid {
innerRVectors = []; // list of downright, upright, up, upleft, downleft, down vectors
offsetModifiers = [
[ 1, (column) => column % 2 == 0? 0: 1 ], // downright
[ 1, (column) => column % 2 == 0? -1: 0 ], //upright
[ 0, (column) => -1 ], //up
[-1, (column) => column % 2 == 0? -1: 0 ], //upleft
[-1, (column) => column % 2 == 0? 0: 1 ], //downleft
[ 0, (column) => 1 ] //down
];

cubeModifiers = [[1,0,-1],[1,-1, 0],[0,-1,1],[-1,0,1],[-1,1,0],[0,1,-1]];

iterator = null;

currentOffset = [0,0];
currentIndex = 0;
currentCube = [0,0,0];
currentDouble = [0,0];
currentRing = 0;

constructor(edgeSize, spiral = false, rotation = 0, clockwise = false) {
this.edgeSize = edgeSize;
this.innerSize = Math.sqrt(.75) * edgeSize;
this.spiral = spiral;
this.rotation = rotation;
this.clockwise = clockwise
}
setInnerRVectors() {
for(let i = 1; i <= 6; i++) {
this.innerRVectors.push(this.trans2(this.rot2(((Math.PI / 3) * i * (this.clockwise? -1: 1)) - this.rotation), [0, this.innerSize * 2]));
}
}
updateCoordination(index) {
this.currentDouble = [this.currentOffset[0], this.currentOffset[1] + this.currentOffset[1] + (this.currentOffset[0] % 2 == 0? 0:1)];
this.currentIndex++;
}
*spiralCellPositions() {
let position = turtle.pos();

yield this.yieldCell(position);
while(this.currentRing++ !== false) {
this.updateCoordination(5);
yield this.yieldCell(position);

for(let j = 0; j < 6; j++) {
for(let i = 0; i < this.currentRing - (j == 0? 1: 0); i++) {

this.updateCoordination(j);

yield this.yieldCell(position);
}
}
}
}
*ringCellPositions() {
let position = turtle.pos();

yield this.yieldCell(position);
while(this.currentRing++ !== false) {
let rPosition = this.add2(position, this.scale2(this.innerRVectors[5], this.currentRing));

this.currentOffset = [0, this.currentRing];
this.currentDouble = [this.currentOffset[0], this.currentOffset[1] + this.currentOffset[1] + (this.currentOffset[0] % 2 == 0? 0:1)];
this.currentCube = [0, this.currentRing, -this.currentRing];
this.currentIndex++;

yield this.yieldCell(rPosition);
for(let i = 1; i <= 6; i++) {
for(let n = 0; n < (i == 6? this.currentRing - 1: this.currentRing); n++) {
rPosition = this.add2(rPosition, this.innerRVectors[i % 6]);
this.updateCoordination(i % 6);
yield this.yieldCell(rPosition);
}
}
}
}
yieldCell(position) {
return new HexCell(this.edgeSize, position, this.currentIndex, this.currentOffset, this.currentCube, this.currentDouble, this.currentRing, this.rotation);
}
nextCell() {
if(this.iterator == null) {
this.setInnerRVectors();
this.iterator = this.spiral? this.spiralCellPositions(): this.ringCellPositions();
this.index = 0;
}
return this.iterator.next().value;
}
nthTriangular(n) { return ((n * n) + n) / 2; }

//
// Vector math
//

rot2(a) { return [Math.cos(a), -Math.sin(a), Math.sin(a), Math.cos(a)]; }
trans2(m, a) { return [m[0]*a[0]+m[2]*a[1], m[1]*a[0]+m[3]*a[1]]; }
scale2(a,b) { return [a[0]*b,a[1]*b]; }
}

class HexCell {
constructor(size, position, index, offset, cube, double, ring, rotation) {
this.position = position;
this.index = index;
this.offset = offset;
this.cube = cube;
this.double = double;
this.ring = ring;
this.size = size;
this.rotation = rotation;
}
getBorderPoints(turtle, borderSize = null) {
let isDown = turtle.isdown();
turtle.up();

if(borderSize == null) {
borderSize = this.size;
}

turtle.forward(borderSize);
turtle.right(turtle._fullCircle / 3);

let points = [];
for(let i = 0; i < 6; i++) {
points.push(turtle.pos());
turtle.forward(borderSize);
turtle.right(turtle._fullCircle / 6);
}

turtle.left(turtle._fullCircle / 3);
turtle.forward(-borderSize);

if(isDown) { turtle.down(); }

return points;
}
drawBorder(turtle, borderSize = null) {
let points = this.getBorderPoints(turtle, borderSize);
turtle.jump(points[0]);
for(let i = 1; i < points.length; i++) {
turtle.goto(points[i]);
}
turtle.goto(points[0]);
turtle.jump(this.position);
}
}

let hg = new HexGrid(    ((height + gap) / hsq3) * (edgeCubes + 1) / 2   );
```