hex-relations.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "A0 = {(0, 0, 0)}\n",
    "A1 = set()\n",
    "B0 = set()\n",
    "B1 = set()\n",
    "\n",
    "\n",
    "class InvalidPointError(ValueError):\n",
    "    \"\"\"Point is not in the lattice.\"\"\"\n",
    "\n",
    "\n",
    "def neighbors_Ic(v):\n",
    "    sum_ = sum(v)\n",
    "    if sum_ % 12 == 0:\n",
    "        if v[0] % 6 + v[1] % 6 + v[2] % 6 > 0:\n",
    "            raise InvalidPointError(v)\n",
    "        return {\n",
    "            (v[0] - 3, v[1] - 3, v[2] - 3),\n",
    "            (v[0] - 3, v[1] + 3, v[2] + 3),\n",
    "            (v[0] + 3, v[1] - 3, v[2] + 3),\n",
    "            (v[0] + 3, v[1] + 3, v[2] - 3),\n",
    "        }\n",
    "    elif sum_ % 12 == 3:\n",
    "        if (v[0] - 3) % 6 + (v[1] - 3) % 6 + (v[2] - 3) % 6 > 0:\n",
    "            raise InvalidPointError(v)\n",
    "        return {\n",
    "            (v[0] + 3, v[1] + 3, v[2] + 3),\n",
    "            (v[0] + 3, v[1] - 3, v[2] - 3),\n",
    "            (v[0] - 3, v[1] + 3, v[2] - 3),\n",
    "            (v[0] - 3, v[1] - 3, v[2] + 3),\n",
    "        }\n",
    "    else:\n",
    "        raise InvalidPointError(v)\n",
    "\n",
    "\n",
    "def neighbors_Ih(v):\n",
    "    sum_ = sum(v)\n",
    "    if sum_ % 3 != 0:\n",
    "        raise InvalidPointError(v)\n",
    "    s = sum_ // 3\n",
    "    if s % 8 == 0:\n",
    "        if (v[0] - s) % 6 + (v[1] - s) % 6 + (v[2] - s) % 6 > 0:\n",
    "            raise InvalidPointError(v)\n",
    "        return {\n",
    "            (v[0] - 3, v[1] - 3, v[2] - 3),\n",
    "            (v[0] - 3, v[1] + 3, v[2] + 3),\n",
    "            (v[0] + 3, v[1] - 3, v[2] + 3),\n",
    "            (v[0] + 3, v[1] + 3, v[2] - 3),\n",
    "        }\n",
    "    elif s % 8 == 1:\n",
    "        if (v[0] - s - 2) % 6 + (v[1] - s - 2) % 6 + (v[2] - s - 2) % 6 > 0:\n",
    "            raise InvalidPointError(v)\n",
    "        return {\n",
    "            (v[0] + 3, v[1] + 3, v[2] + 3),\n",
    "            (v[0] + 3, v[1] - 3, v[2] - 3),\n",
    "            (v[0] - 3, v[1] + 3, v[2] - 3),\n",
    "            (v[0] - 3, v[1] - 3, v[2] + 3),\n",
    "        }\n",
    "    elif s % 8 == 4:\n",
    "        if (v[0] - s - 2) % 6 + (v[1] - s - 2) % 6 + (v[2] - s - 2) % 6 > 0:\n",
    "            raise InvalidPointError(v)\n",
    "        return {\n",
    "            (v[0] - 3, v[1] - 3, v[2] - 3),\n",
    "            (v[0] + 5, v[1] - 1, v[2] - 1),\n",
    "            (v[0] - 1, v[1] + 5, v[2] - 1),\n",
    "            (v[0] - 1, v[1] - 1, v[2] + 5),\n",
    "        }\n",
    "    elif s % 8 == 5:\n",
    "        if (v[0] - s) % 6 + (v[1] - s) % 6 + (v[2] - s) % 6 > 0:\n",
    "            raise InvalidPointError(v)\n",
    "        return {\n",
    "            (v[0] + 3, v[1] + 3, v[2] + 3),\n",
    "            (v[0] - 5, v[1] + 1, v[2] + 1),\n",
    "            (v[0] + 1, v[1] - 5, v[2] + 1),\n",
    "            (v[0] + 1, v[1] + 1, v[2] - 5),\n",
    "        }\n",
    "    else:\n",
    "        raise InvalidPointError(v)\n",
    "\n",
    "\n",
    "def neighbors_A0(v):\n",
    "    return dict(\n",
    "        A0=set(),\n",
    "        A1=set(),\n",
    "        B1={(v[0] + 3, v[1] + 3, v[2] + 3)},\n",
    "        B0={\n",
    "            (v[0] + 3, v[1] - 3, v[2] - 3),\n",
    "            (v[0] - 3, v[1] + 3, v[2] - 3),\n",
    "            (v[0] - 3, v[1] - 3, v[2] + 3),\n",
    "        },\n",
    "    )\n",
    "\n",
    "\n",
    "def neighbors_B0(v):\n",
    "    return dict(\n",
    "        B1=set(),\n",
    "        B2=set(),\n",
    "        A1={(v[0] - 3, v[1] - 3, v[2] - 3)},\n",
    "        A0={\n",
    "            (v[0] - 3, v[1] + 3, v[2] + 3),\n",
    "            (v[0] + 3, v[1] - 3, v[2] + 3),\n",
    "            (v[0] + 3, v[1] + 3, v[2] - 3),\n",
    "        },\n",
    "    )\n",
    "\n",
    "\n",
    "def neighbors_A1(v):\n",
    "    return dict(\n",
    "        A0=set(),\n",
    "        A1=set(),\n",
    "        B0={(v[0] + 3, v[1] + 3, v[2] + 3)},\n",
    "        B1={\n",
    "            (v[0] - 5, v[1] + 1, v[2] + 1),\n",
    "            (v[0] + 1, v[1] - 5, v[2] + 1),\n",
    "            (v[0] + 1, v[1] + 1, v[2] - 5),\n",
    "        },\n",
    "    )\n",
    "\n",
    "\n",
    "def neighbors_B1(v):\n",
    "    return dict(\n",
    "        B1=set(),\n",
    "        B2=set(),\n",
    "        A0={(v[0] - 3, v[1] - 3, v[2] - 3)},\n",
    "        A1={\n",
    "            (v[0] + 5, v[1] - 1, v[2] - 1),\n",
    "            (v[0] - 1, v[1] + 5, v[2] - 1),\n",
    "            (v[0] - 1, v[1] - 1, v[2] + 5),\n",
    "        },\n",
    "    )\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 770,
   "metadata": {},
   "outputs": [],
   "source": [
    "facet_normals_ih = [\n",
    "    (6, 6, 6),\n",
    "    (11, -1, -1),\n",
    "    (-1, 11, -1),\n",
    "    (-1, -1, 11),\n",
    "    (-5, 7, 7),\n",
    "    (7, -5, 7),\n",
    "    (7, 7, -5),\n",
    "]\n",
    "facet_normals_ih += [(-n[0], -n[1], -n[2]) for n in facet_normals_ih]\n",
    "def ih_graph_distance(v):\n",
    "    dots = [n[0] * v[0] + n[1] * v[1] + n[2] * v[2] for n in facet_normals_ih]\n",
    "    for dot in dots:\n",
    "        assert dot % 3 == 0\n",
    "    dot = max(dots) // 3\n",
    "    q, r = divmod(dot + 4, 24)\n",
    "    res = 2 * q\n",
    "    if r >= 9:\n",
    "        res += 1\n",
    "    return res\n",
    "\n",
    "facet_normals_ic = [\n",
    "    (2, 0, 0),\n",
    "    (0, 2, 0),\n",
    "    (0, 0, 2),\n",
    "    (1, 1, 1),\n",
    "    (-1, 1, 1),\n",
    "    (1, -1, 1),\n",
    "    (1, 1, -1),\n",
    "]\n",
    "facet_normals_ic += [(-n[0], -n[1], -n[2]) for n in facet_normals_ic]\n",
    "def ic_graph_distance(v):\n",
    "    dots = [n[0] * v[0] + n[1] * v[1] + n[2] * v[2] for n in facet_normals_ic]\n",
    "    for dot in dots:\n",
    "        assert dot % 3 == 0\n",
    "    dot = max(dots) // 3\n",
    "    q, r = divmod(dot, 4)\n",
    "    res = 2 * q\n",
    "    if r >= 1:\n",
    "        res += 1\n",
    "    return res\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 771,
   "metadata": {},
   "outputs": [],
   "source": [
    "S = {(0, 0, 0)}\n",
    "lens = [1]\n",
    "boundary = S\n",
    "facet_counts = []\n",
    "facet_vals = []\n",
    "facet_vals2 = []\n",
    "facet_counts2 = []\n",
    "facet_vals3 = []\n",
    "facet_counts3 = []\n",
    "thresholds = []"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 787,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[1, 4, 12, 24, 42, 64, 92, 124, 162, 204, 252, 304, 362, 424, 492, 564, 642]\n",
      "{16}\n",
      "thresholds=[1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7, 7, 8, 8, 9, 9, 10, 10, 11, 11, 12, 12, 13, 13, 14, 14, 15, 15, 16, 16]\n",
      "facet_vals=[9, 12, 21, 24, 33, 36, 45, 48, 57, 60, 69, 72, 81, 84, 93, 96]\n",
      "[1, 3, 3, 6, 6, 10, 10, 15, 15, 21, 21, 28, 28, 36, 36, 45]\n",
      "facet_vals2=[-3, 0, 9, 12, 21, 24, 33, 36, 45, 48, 57, 60, 69, 72, 81, 84]\n",
      "[3, 6, 6, 9, 9, 12, 12, 15, 15, 18, 18, 21, 21, 24, 24, 27]\n",
      "[-1000, -12, -3, 0, 9, 12, 21, 24, 33, 36, 45, 48, 57, 60, 69, 72]\n",
      "[0, 3, 9, 12, 12, 15, 15, 18, 18, 21, 21, 24, 24, 27, 27, 30]\n"
     ]
    }
   ],
   "source": [
    "boundary_neighbors = set.union(*[neighbors_Ic(v) for v in boundary])\n",
    "boundary = boundary_neighbors - S\n",
    "S |= boundary\n",
    "# print(to_mma(S))\n",
    "# print(S)\n",
    "lens.append(len(boundary))\n",
    "print(lens)\n",
    "vals = [ic_graph_distance(v) for v in boundary]\n",
    "print(set(vals))\n",
    "thresholds += [min(vals), max(vals)]\n",
    "print(f\"{thresholds=}\")\n",
    "# fourty_seven = [v for v in boundary if ic_graph_distance(v) == 47]\n",
    "# fourty_four = [v for v in boundary if ic_graph_distance(v) == 44]\n",
    "# print(f\"{fourty_seven=}\")\n",
    "# print(f\"{fourty_four=}\")\n",
    "\n",
    "\n",
    "dots = sorted([-1 * v[0] + 1 * v[1] + 1 * v[2] for v in boundary])\n",
    "# print([v for v in S if v[0] + v[1] + v[2] == -9])\n",
    "facet_val = sorted(set(dots))[-1]\n",
    "facet_val2 = sorted(set(dots))[-2]\n",
    "if len(set(dots)) == 2:\n",
    "    facet_val3 = -1000\n",
    "else:\n",
    "    facet_val3 = sorted(set(dots))[-3]\n",
    "facet_count = len([None for n in dots if n == facet_val])\n",
    "facet_count2 = len([None for n in dots if n == facet_val2])\n",
    "facet_count3 = len([None for n in dots if n == facet_val3])\n",
    "facet_vals.append(facet_val)\n",
    "facet_counts.append(facet_count)\n",
    "facet_vals2.append(facet_val2)\n",
    "facet_counts2.append(facet_count2)\n",
    "facet_vals3.append(facet_val3)\n",
    "facet_counts3.append(facet_count3)\n",
    "\n",
    "\n",
    "print(f\"{facet_vals=}\")\n",
    "print(facet_counts)\n",
    "print(f\"{facet_vals2=}\")\n",
    "print(facet_counts2)\n",
    "print(facet_vals3)\n",
    "print(facet_counts3)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 624,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "{-81,\n",
       " -77,\n",
       " -72,\n",
       " -68,\n",
       " -57,\n",
       " -53,\n",
       " -48,\n",
       " -44,\n",
       " -33,\n",
       " -29,\n",
       " -24,\n",
       " -20,\n",
       " -9,\n",
       " -5,\n",
       " 0,\n",
       " 4,\n",
       " 15,\n",
       " 19,\n",
       " 24,\n",
       " 28,\n",
       " 39,\n",
       " 43,\n",
       " 48,\n",
       " 52,\n",
       " 63,\n",
       " 67,\n",
       " 72,\n",
       " 76,\n",
       " 87,\n",
       " 91}"
      ]
     },
     "execution_count": 624,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "dots = sorted([(-5 * v[0] + 7 * v[1] + 7 * v[2]) // 3 for v in S])\n",
    "set(dots)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 621,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "{-270, -216, -198, -144, -126, -72, -54, 0, 18, 72, 90, 144, 162, 216, 234}"
      ]
     },
     "execution_count": 621,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "set(dots)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 469,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19]\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "[0, 0, 0, 0, 6, 6, 6, 6, 6, 12, 12, 12, 12, 12, 12, 12, 18, 18, 18, 18]"
      ]
     },
     "execution_count": 469,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "def even_round(n: int, m: int) -> int:\n",
    "    \"\"\"For even m, round to the nearest multiple of m.\n",
    "    \n",
    "    Ties are rounded to the nearest multiple of 2m.\n",
    "    \"\"\"\n",
    "    assert m % 2 == 0\n",
    "    half_m = m // 2\n",
    "    res = (n + half_m) // m\n",
    "    if n % (2 * m) == half_m:\n",
    "        pass\n",
    "        res -= 1\n",
    "    return res * m\n",
    "print(list(range(20)))\n",
    "[even_round(n, 6) for n in range(20)]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 534,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": 480,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[[-54, -27, -27, -27, -27, -27, -27, 54, 27, 27, 27, 27, 27, 27],\n",
       " [18, 33, 33, -39, -15, -15, 57, -18, -33, -33, 39, 15, 15, -57],\n",
       " [18, -39, 33, 33, 57, -15, -15, -18, 39, -33, -33, -57, 15, 15],\n",
       " [18, 33, -39, 33, -15, 57, -15, -18, -33, 39, -33, 15, -57, 15]]"
      ]
     },
     "execution_count": 480,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "[[n[0] * v[0] + n[1] * v[1] + n[2] * v[2] for n in facet_normals] for v in boundary]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "[1, 4, 12, 25, 44]\n",
    "[(3, -9, -3), (-3, -9, 3), (-3, -3, -3), (3, -3, -9), (-9, -3, 3), (-3, 3, -9), (-9, 3, -3)]\n",
    "facet_vals=[9, 12, 21, 24]\n",
    "[1, 3, 3, 7]\n",
    "facet_vals2=[0, 9, 12, 21]\n",
    "[1, 1, 3, 3]\n",
    "[-3, 0, 9, 12]\n",
    "[3, 7, 7, 12]\n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 286,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[3,\n",
       " 4,\n",
       " 5,\n",
       " 8,\n",
       " 10,\n",
       " 14,\n",
       " 16,\n",
       " 21,\n",
       " 24,\n",
       " 30,\n",
       " 33,\n",
       " 40,\n",
       " 44,\n",
       " 52,\n",
       " 56,\n",
       " 65,\n",
       " 70,\n",
       " 80,\n",
       " 85,\n",
       " 96,\n",
       " 102,\n",
       " 114,\n",
       " 120,\n",
       " 133,\n",
       " 140,\n",
       " 154,\n",
       " 161,\n",
       " 176,\n",
       " 184,\n",
       " 200,\n",
       " 208,\n",
       " 225,\n",
       " 234,\n",
       " 252,\n",
       " 261,\n",
       " 280,\n",
       " 290,\n",
       " 310,\n",
       " 320]"
      ]
     },
     "execution_count": 286,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "[c1+c2 for c1, c2 in zip(facet_counts, facet_counts2)]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 656,
   "metadata": {},
   "outputs": [],
   "source": [
    "dot_freq = []\n",
    "for a in range(-12,12):\n",
    "    for b in range(-12,12):\n",
    "        for c in range(-12, 12):\n",
    "            if gcd(a, b, c) == 1:\n",
    "                dots = sorted([a * v[0] + b * v[1] + c * v[2] for v in boundary])\n",
    "                facet_val = sorted(dots)[-1]\n",
    "                facet_count = len([None for n in dots if n == facet_val])\n",
    "                dot_freq.append((facet_count, a, b, c))\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 657,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[(15, 11, -1, -1),\n",
       " (15, 11, 1, 1),\n",
       " (15, 11, 2, 2),\n",
       " (15, 11, 3, 3),\n",
       " (15, 11, 4, 4),\n",
       " (15, 11, 5, 5),\n",
       " (15, 11, 6, 6),\n",
       " (15, 11, 7, 7),\n",
       " (15, 11, 8, 8),\n",
       " (15, 11, 9, 9),\n",
       " (15, 11, 10, 10),\n",
       " (105, -1, -1, -1),\n",
       " (105, -1, 1, 1),\n",
       " (105, 1, -1, 1),\n",
       " (105, 1, 1, -1),\n",
       " (120, -1, -1, 1),\n",
       " (120, -1, 1, -1),\n",
       " (120, 1, -1, -1),\n",
       " (120, 1, 1, 1),\n",
       " (210, -1, 0, 0),\n",
       " (210, 0, -1, 0),\n",
       " (210, 0, 0, -1),\n",
       " (210, 0, 0, 1),\n",
       " (210, 0, 1, 0),\n",
       " (210, 1, 0, 0)]"
      ]
     },
     "execution_count": 657,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sorted(dot_freq)[-25:]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 105,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "6"
      ]
     },
     "execution_count": 105,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "from math import gcd\n",
    "gcd(12, 18)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 395,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[0, 18, 72, 90, 144, 162, 216, 234, 288, 306, 360, 378, 432, 450, 504]\n",
      "[1, 3, 3, 7, 7, 12, 12, 19, 19, 27, 27, 37, 37, 48, 48]\n",
      "[-72, -54, 0, 18, 72, 90, 144, 162, 216, 234, 288, 306, 360, 378, 432]\n"
     ]
    }
   ],
   "source": [
    "from math import ceil\n",
    "\n",
    "# # v[0] + v[1] + v[2]\n",
    "# print([6 * n + - 3 * (n % 2) for n in range(15)])\n",
    "# print([ceil(((n + 3)//2)**2 *3/4) for n in range(15)])\n",
    "\n",
    "# # facet_vals2=[-9, 0, 3, 12, 15, 24, 27, 36, 39, 48, 51, 60]\n",
    "# print([6 * n - 12 + - 3 * (n % 2) for n in range(15)])\n",
    "\n",
    "\n",
    "# 6 * v[0] + 6 * v[1] + 6 * v[2]\n",
    "print([36 * n - 18 * (n % 2) for n in range(15)])\n",
    "print([ceil(((n + 3)//2)**2 *3/4) for n in range(15)])\n",
    "\n",
    "# facet_vals2=[-9, 0, 3, 12, 15, 24, 27, 36, 39, 48, 51, 60]\n",
    "print([36 * n - 72 - 18 * (n % 2) for n in range(15)])\n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 396,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[0, 54, 72, 126, 144, 198, 216, 270, 288, 342, 360, 414, 432, 486, 504]\n",
      "[1, 1, 3, 3, 7, 7, 12, 12, 19, 19, 27, 27, 37, 37, 48]\n",
      "[-72, -18, 0, 54, 72, 126, 144, 198, 216, 270, 288, 342, 360, 414, 432]\n"
     ]
    }
   ],
   "source": [
    "# # -v[0] - v[1] - v[2]\n",
    "\n",
    "# print([6 * n + 3 * (n % 2) for n in range(15)])\n",
    "# print([ceil(((n + 2)//2)**2 *3/4) for n in range(15)])\n",
    "\n",
    "# # facet_vals2=[-3, 0, 9, 12, 21, 24, 33, 36, 45]\n",
    "# print([6 * n - 12 + + 3 * (n % 2) for n in range(15)])\n",
    "\n",
    "\n",
    "# -6 * v[0] - 6 * v[1] - 6 * v[2]\n",
    "\n",
    "print([36 * n + 18 * (n % 2) for n in range(15)])\n",
    "print([ceil(((n + 2)//2)**2 *3/4) for n in range(15)])\n",
    "\n",
    "# facet_vals2=[-3, 0, 9, 12, 21, 24, 33, 36, 45]\n",
    "print([36 * n - 72 + 18 * (n % 2) for n in range(15)])\n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 307,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[0, 45, 72, 117, 144, 189, 216, 261, 288, 333, 360, 405, 432, 477, 504]\n",
      "[-12, 33, 60, 105, 132, 177, 204, 249, 276, 321, 348, 393, 420, 465, 492]\n",
      "[-72, -27, 0, 45, 72, 117, 144, 189, 216, 261, 288, 333, 360, 405, 432]\n"
     ]
    }
   ],
   "source": [
    "# 11 * v[0] - v[1] - v[2]\n",
    "\n",
    "print([36 * n + 9 * (n % 2) for n in range(15)])\n",
    "print([36 * n - 12 + 9 * (n % 2) for n in range(15)])\n",
    "\n",
    "print([36 * n - 72 + 9 * (n % 2) for n in range(15)])\n",
    "\n",
    "# facet_vals=[33, 72, 117, 144, 189, 216, 261, 288, 333, 360, 405, 432, 477, 504, 549, 576, 621, 648, 693, 720, 765, 792, 837, 864, 909, 936, 981, 1008, 1053, 1080, 1125, 1152, 1197, 1224, 1269, 1296, 1341, 1368, 1413, 1440]\n",
    "# [2, 2, 2, 5, 3, 7, 7, 12, 9, 15, 15, 22, 18, 26, 26, 35, 30, 40, 40, 51, 45, 57, 57, 70, 63, 77, 77, 92, 84, 100, 100, 117, 108, 126, 126, 145, 135, 155, 155, 176]\n",
    "# facet_vals2=[-27, 60, 105, 132, 177, 204, 249, 276, 321, 348, 393, 420, 465, 492, 537, 564, 609, 636, 681, 708, 753, 780, 825, 852, 897, 924, 969, 996, 1041, 1068, 1113, 1140, 1185, 1212, 1257, 1284, 1329, 1356, 1401, 1428]\n",
    "# A211539:\n",
    "# [1, 2, 3, 3, 7, 7, 9, 9, 15, 15, 18, 18, 26, 26, 30, 30, 40, 40, 45, 45, 57, 57, 63, 63, 77, 77, 84, 84, 100, 100, 108, 108, 126, 126, 135, 135, 155, 155, 165, 165]\n",
    "\n",
    "# -39, 0, 45, 72, 117, 144, 189, 216, 261, 288, 333, 360, 405, 432, 477, 504,\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 346,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[12, 39, 84, 111, 156, 183, 228, 255, 300, 327, 372, 399, 444, 471, 516]\n",
      "[0, 27, 72, 99, 144, 171, 216, 243, 288, 315, 360, 387, 432, 459, 504]\n",
      "[-60, -33, 12, 39, 84, 111, 156, 183, 228, 255, 300, 327, 372, 399, 444]\n"
     ]
    }
   ],
   "source": [
    "# -11 * v[0] + v[1] + v[2]\n",
    "\n",
    "print([36 * n + 12 - 9 * (n % 2) for n in range(15)])\n",
    "print([36 * n - 9 * (n % 2) for n in range(15)])\n",
    "print([36 * n - 60 - 9 * (n % 2) for n in range(15)])\n",
    "\n",
    "# facet_vals=[39, 84, 111, 156, 183, 228, 255, 300, 327]\n",
    "# [1, 1, 3, 2, 5, 5, 9, 7, 12]\n",
    "# facet_vals2=[27, 72, 99, 144, 171, 216, 243, 288, 315]\n",
    "# [1, 2, 2, 5, 5, 7, 7, 12, 12]\n",
    "# [-33, 12, 39, 84, 111, 156, 183, 228, 255]\n",
    "# [2, 3, 4, 4, 4, 8, 7, 8, 6]\n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 368,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[12, 57, 84, 129, 156, 201, 228, 273, 300, 345, 372, 417, 444, 489, 516]\n",
      "[0, 45, 72, 117, 144, 189, 216, 261, 288, 333, 360, 405, 432, 477, 504]\n",
      "[-60, -15, 12, 57, 84, 129, 156, 201, 228, 273, 300, 345, 372, 417, 444]\n"
     ]
    }
   ],
   "source": [
    "# -5 * v[0] + 7 * v[1] + 7 * v[2]\n",
    "\n",
    "print([36 * n + 12 + 9 * (n % 2) for n in range(15)])\n",
    "print([36 * n + 9 * (n % 2) for n in range(15)])\n",
    "print([36 * n - 60 + 9 * (n % 2) for n in range(15)])\n",
    "\n",
    "# facet_vals=[57, 84, 129, 156, 201, 228, 273, 300, 345]\n",
    "# [1, 1, 2, 2, 5, 5, 7, 7, 12]\n",
    "# facet_vals2=[-15, 72, 117, 144, 189, 216, 261, 288, 333]\n",
    "# [2, 2, 2, 5, 3, 7, 7, 12, 9]\n",
    "# [-27, 12, 57, 84, 129, 156, 201, 228, 273]\n",
    "# [1, 3, 2, 4, 5, 8, 4, 8, 8]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 382,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[0, 27, 72, 99, 144, 171, 216, 243, 288, 315, 360, 387, 432, 459, 504]\n",
      "[-12, 15, 60, 87, 132, 159, 204, 231, 276, 303, 348, 375, 420, 447, 492]\n",
      "[-72, -45, 0, 27, 72, 99, 144, 171, 216, 243, 288, 315, 360, 387, 432]\n"
     ]
    }
   ],
   "source": [
    "# 5 * v[0] - 7 * v[1] - 7 * v[2]\n",
    "\n",
    "print([36 * n - 9 * (n % 2) for n in range(15)])\n",
    "print([36 * n - 12 - 9 * (n % 2) for n in range(15)])\n",
    "print([36 * n - 72 - 9 * (n % 2) for n in range(15)])\n",
    "\n",
    "# facet_vals=[27, 72, 99, 144, 171, 216, 243, 288, 315, 360]\n",
    "# [1, 2, 2, 5, 5, 7, 7, 12, 12, 15]\n",
    "# facet_vals2=[15, 60, 87, 132, 159, 204, 231, 276, 303, 348]\n",
    "# [2, 2, 5, 3, 7, 7, 12, 9, 15, 15]\n",
    "# [-57, 0, 27, 72, 99, 144, 171, 216, 243, 288]\n",
    "# [1, 2, 4, 5, 8, 4, 8, 8, 13, 6]\n",
    "\n",
    "\n",
    "# facet_vals=[27, 72, 99, 144, 171, 216, 243, 288, 315, 360]\n",
    "# [1, 2, 2, 5, 5, 7, 7, 12, 12, 15]\n",
    "# facet_vals2=[15, 60, 87, 132, 159, 204, 231, 276, 303, 348]\n",
    "# [2, 2, 5, 3, 7, 7, 12, 9, 15, 15]\n",
    "# [0, 27, 72, 99, 144, 171, 216, 243, 288, 315]\n",
    "# [1, 1, 2, 2, 5, 5, 7, 7, 12, 12]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {},
   "outputs": [],
   "source": [
    "for x in range(-20,20):\n",
    "    for y in range(-20, 20):\n",
    "        for z in range(-20, 20):\n",
    "            if (x,y,z) in S:\n",
    "                continue\n",
    "            try:\n",
    "                neighbors_Ih((x,y,z))\n",
    "            except InvalidPointError:\n",
    "                continue\n",
    "            print((x,y,z))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 134,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "{(16, 15, 17), (16, 21, 11), (22, 15, 11), (22, 21, 17)}"
      ]
     },
     "execution_count": 134,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "neighbors((19, 18, 14))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "A0: {(20, -4, 8), (22, 4, 22), (-6, 24, -18), (10, -8, -26), (-20, -2, -2), (10, 10, 28), (-22, -10, -16), (-8, 10, -26), (-18, 6, 12), (-4, 2, 26), (-12, -6, 18), (-4, 26, 2), (-2, -26, 4), (6, 6, -12), (8, 8, 8), (2, 8, 14), (10, 16, 22), (16, 4, 28), (-24, 12, 12), (-12, 0, 12), (16, 28, 4), (24, -24, 0), (18, -24, 6), (22, 22, 4), (24, 0, -24), (28, 10, 10), (-6, 12, -6), (0, 0, 0), (10, -20, -14), (20, 14, -10), (12, 6, -18), (14, 14, -4), (-20, -14, 10), (-22, -22, -4), (-14, 10, -20), (-8, -2, -14), (-18, -6, 24), (-4, 14, 14), (12, 12, -24), (6, -6, 0), (8, -4, 20), (2, -4, 26), (8, 20, -4), (10, 28, 10), (-12, -12, 24), (-16, -4, -28), (-10, -16, -22), (22, 10, 16), (24, -12, -12), (2, 26, -4), (-6, 0, 6), (0, -12, 12), (4, 22, 22), (20, 2, 2), (12, -6, -6), (14, 2, 8), (-6, 6, 0), (-14, -2, -8), (-10, 8, 26), (-8, -14, -2), (4, 28, 16), (26, 8, -10), (6, 12, -18), (12, 0, -12), (4, -8, -20), (-26, 4, -2), (18, -18, 0), (-12, 24, -12), (-16, -16, -16), (-10, -28, -10), (-2, 4, -26), (0, 6, -6), (2, 14, 8), (-6, -12, 18), (0, -24, 24), (14, 20, -10), (20, -10, 14), (12, -18, 6), (14, -10, 20), (-8, 4, -20), (-14, -14, 4), (-8, -26, 10), (-6, 18, -12), (-4, 20, 8), (-10, 20, 14), (4, 16, 28), (26, -4, 2), (6, 0, -6), (12, -12, 0), (4, -20, -8), (-18, 0, 18), (-24, 0, 24), (-26, -8, 10), (-18, 24, -6), (-12, 12, 0), (-16, -28, -4), (-2, -8, -14), (0, -6, 6), (2, 2, 20), (-4, -16, -28), (8, 26, -10), (0, 18, -18), (-26, -2, 4), (10, -26, -8), (14, 8, 2), (6, -18, 12), (-10, -10, -28), (-8, -8, -8), (0, 24, -24), (-4, 8, 20), (6, -12, 6), (12, -24, 12), (-14, 4, -14), (-20, 4, -8), (-24, 18, 6), (-22, -4, -22), (-18, 12, 6), (18, 6, -24), (4, -2, -26), (-6, -6, 12), (-2, -20, -2), (0, -18, 18), (-4, -28, -16), (8, 14, 2), (-18, 18, 0), (-26, 10, -8), (10, -14, -20), (14, -4, 14), (-16, -10, -22), (-14, -8, -2), (-20, -8, 4), (-8, -20, 4), (28, 16, 4), (-10, -22, -16), (-4, -22, -22), (0, 12, -12), (6, -24, 18), (-28, -4, -16), (16, 22, 10), (-24, 6, 18), (-22, -16, -10), (-12, 18, -6), (-10, 26, 8), (26, 2, -4), (18, -6, -12), (24, -6, -18), (-2, -2, -20), (4, -14, -14), (-6, -18, 24), (8, 2, 14), (18, 0, -18), (20, 8, -4), (22, 16, 10), (28, 4, 16), (-14, -20, 10), (-16, -22, -10), (-20, 10, -14), (10, 22, 16), (-28, -16, -4), (16, 10, 22), (-12, 6, 6), (-10, 14, 20), (26, -10, 8), (24, -18, -6), (-2, -14, -8), (4, -26, -2), (6, 18, -24), (2, 20, 2), (16, 16, 16), (-28, -10, -10), (8, -10, 26), (-24, 24, 0), (18, -12, -6)}\n",
      "A1: {(-24, 6, 6), (16, 22, -2), (-12, 18, -18), (-10, 26, -4), (26, 2, -16), (18, -6, -24), (4, -14, -26), (-6, -18, 12), (8, 2, 2), (4, 4, 28), (14, -16, 14), (20, 8, -16), (22, 16, -2), (28, 4, 4), (6, -12, -6), (10, 22, 4), (16, 10, 10), (-24, -6, 18), (-16, 26, 2), (-12, 6, -6), (-10, 14, 8), (26, -10, -4), (-2, -14, -20), (4, -26, -14), (2, 20, -10), (16, 16, 4), (8, -10, 14), (18, -12, -18), (20, -4, -4), (22, 4, 10), (0, 12, -24), (6, -24, 6), (-20, -2, -14), (10, 10, 16), (16, -2, 22), (-18, 6, 0), (-4, 2, 14), (-12, -6, 6), (-10, 2, 20), (-16, 14, 14), (-4, 26, -10), (-2, -26, -8), (6, 6, -24), (8, 8, -4), (2, 8, 2), (10, 16, 10), (16, 4, 16), (-12, 0, 0), (18, -24, -6), (-14, -20, -2), (-20, -20, 4), (28, 10, -2), (20, -16, 8), (-6, 12, -18), (0, 0, -12), (4, 10, 22), (14, 14, -16), (-20, -14, -2), (10, -2, 28), (-8, -2, -26), (-18, -6, 12), (-16, 2, 26), (-4, -10, 26), (-12, -18, 18), (-4, 14, 2), (6, -6, -12), (8, -4, 8), (2, -4, 14), (10, 4, 22), (8, 20, -16), (10, 28, -2), (28, -2, 10), (2, 26, -16), (-6, 0, -6), (-2, 10, 28), (0, -12, 0), (4, 22, 10), (20, 2, -10), (14, 2, -4), (12, -6, -18), (-6, 6, -12), (-14, -2, -20), (-10, 8, 14), (-8, -14, -14), (-16, 20, 8), (26, -16, 2), (4, 28, 4), (12, 0, -24), (8, -16, 20), (2, -16, 26), (-24, 12, 0), (-26, 4, -14), (18, -18, -12), (2, 14, -4), (0, 6, -18), (-6, -12, 6), (0, -24, 12), (-2, 22, 16), (20, -10, 2), (12, -18, -6), (14, -10, 8), (-16, 8, 20), (-14, -14, -8), (-10, -4, 26), (-8, -26, -2), (-6, 18, -24), (-4, 20, -4), (-10, 20, 2), (4, 16, 16), (-2, 28, 10), (26, -4, -10), (6, 0, -18), (12, -12, -12), (4, -20, -20), (-18, 0, 6), (-24, 0, 12), (-12, -12, 12), (-26, -8, -2), (-12, 12, -12), (22, 10, 4), (-2, -8, -26), (0, -6, -6), (2, 2, 8), (-6, -24, 18), (-26, -2, -8), (14, 8, -10), (6, -18, 0), (-8, -8, -20), (-14, -26, 4), (-4, 8, 8), (-2, 16, 22), (12, -24, 0), (-14, 4, -26), (-20, 4, -20), (-24, 18, -6), (-18, -12, 18), (-18, 12, -6), (22, -2, 16), (-6, -6, 0), (-2, -20, -14), (0, -18, 6), (2, -10, 20), (8, 14, -10), (-26, -14, 4), (-18, 18, -12), (14, -4, 2), (-14, -8, -14), (-20, -8, -8), (-8, -20, -8), (-4, -4, 20)}\n",
      "B0: {(-3, 9, -9), (-5, 1, -23), (-11, -17, 1), (-1, -1, 23), (-21, 9, 9), (1, -11, -17), (15, -15, -3), (7, -23, -11), (19, 19, 7), (-15, -9, 21), (25, 7, 13), (-7, 23, 5), (11, 11, -1), (-23, -5, 1), (7, 25, 13), (-17, -17, 7), (23, -7, 5), (-11, 1, -17), (-3, -3, 3), (-5, -11, -11), (9, 9, -21), (19, 7, 19), (-7, 11, 17), (7, 13, 25), (-1, 5, 17), (-11, -11, -5), (-3, -15, 15), (11, -1, 11), (-5, -23, 1), (3, 9, -15), (7, 19, 19), (-1, 11, 11), (25, 13, 7), (-9, 15, -9), (1, -5, -23), (7, -17, -17), (11, 17, -7), (23, -1, -1), (-15, 21, -9), (13, 25, 7), (-21, -3, 21), (-11, -23, 7), (5, -1, 17), (3, -3, -3), (-21, 3, 15), (15, 3, -21), (19, 13, 13), (21, -9, -15), (-9, 3, 3), (1, -17, -11), (-1, 23, -1), (9, -3, -9), (11, 5, 5), (-9, 9, -3), (-15, 9, 3), (-17, 1, -11), (13, 13, 19), (3, 15, -21), (-3, 15, -15), (9, 3, -15), (5, 23, -7), (3, -15, 9), (17, -1, 5), (13, 19, 13), (15, -9, -9), (21, -21, -3), (-9, -9, 15), (5, 5, 11), (17, -7, 11), (9, -15, 3), (11, -7, 17), (-23, 1, -5), (-9, -3, 9), (-15, -3, 15), (-17, -11, 1), (-9, 21, -15), (-3, -21, 21), (3, 3, -9), (-3, 3, -3), (9, -9, -3), (5, 11, 5), (13, 7, 25), (15, -21, 3), (-11, 7, -23), (-7, 17, 11), (-5, -5, -17), (21, -3, -21), (5, -7, 23), (-15, 15, -3), (-23, -11, 7), (-9, -15, 21), (3, -9, 3), (-3, -9, 9), (5, 17, -1), (17, 5, -1), (9, -21, 9), (-17, 7, -17), (17, 11, -7), (-21, 15, 3), (-1, 17, 5), (-3, 21, -21), (-11, -5, -11), (-7, 5, 23), (-5, -17, -5), (1, -23, -5), (-21, 21, -3), (15, -3, -15), (7, -11, -23), (21, -15, -9), (-15, 3, 9), (3, -21, 15), (-23, 7, -11), (-17, -5, -5), (23, 5, -7)}\n",
      "B1: {(-11, -11, 7), (-5, -23, 13), (11, -1, 23), (3, 9, -3), (-1, 11, 23), (21, 3, -15), (-1, -25, -13), (-13, -25, -1), (1, -5, -11), (-7, -13, -19), (11, 17, 5), (-15, 21, 3), (-17, 13, -11), (-7, -7, -25), (3, -3, 9), (-25, -13, -1), (-25, -7, -7), (15, 3, -9), (21, -9, -3), (1, -17, 1), (5, 17, 11), (17, 5, 11), (9, -3, 3), (11, 5, 17), (13, -5, -23), (-23, 13, -5), (-9, 9, 9), (-17, 1, 1), (-15, 9, 15), (-7, -19, -13), (3, 15, -9), (-3, 15, -3), (9, 3, -3), (5, 23, 5), (3, -15, 21), (17, -1, 17), (-19, -13, -7), (15, -9, 3), (-5, 7, -17), (-25, -1, -13), (5, 5, 23), (13, -17, -11), (9, -15, 15), (-1, -19, -19), (-19, -19, -1), (-23, 1, 7), (-9, -3, 21), (-17, -11, 13), (-9, 21, -3), (1, 1, -17), (3, 3, 3), (-3, 3, 9), (5, 11, 17), (9, -9, 9), (9, 15, -15), (23, 5, 5), (-11, 7, -11), (-5, -5, -5), (15, 9, -15), (7, 1, -23), (11, 11, 11), (21, -3, -9), (-15, 15, 9), (3, -9, 15), (-3, -9, 21), (-17, 7, -5), (17, 11, 5), (-1, 17, 17), (-3, 21, -9), (-5, 13, -23), (-11, -5, 1), (-13, -13, -13), (-5, -17, 7), (1, -23, 7), (15, -3, -3), (7, -11, -11), (21, -15, 3), (13, -11, -17), (23, 11, -1), (-13, -7, -19), (-19, -7, -13), (-15, 3, 21), (-7, -25, -7), (1, 7, -23), (7, -5, -17), (-23, 7, 1), (-17, -5, 7), (17, 17, -1), (-11, 13, -17), (-3, 9, 3), (-5, 1, -11), (-11, -17, 13), (11, 23, -1), (1, -11, -5), (15, -15, 9), (13, -23, -5), (23, -1, 11), (7, -23, 1), (-13, -19, -7), (-9, 15, 3), (7, -17, -5), (-13, -1, -25), (-19, -1, -19), (-23, -5, 13), (-11, 1, -5), (-5, -11, 1), (-3, -3, 15), (3, 21, -15), (9, 9, -9), (-1, 23, 11), (-1, -13, -25), (-9, 3, 15)}\n"
     ]
    }
   ],
   "source": [
    "\n",
    "nA0 = [neighbors_A0(a0) for a0 in A0]\n",
    "nB0 = [neighbors_B0(b0) for b0 in B0]\n",
    "nA1 = [neighbors_A1(a1) for a1 in A1]\n",
    "nB1 = [neighbors_B1(b1) for b1 in B1]\n",
    "newA0 = set.union(set(), *[n[\"A0\"] for n in nB0]) | set.union(set(), *[n[\"A0\"] for n in nB1])\n",
    "newA1 = set.union(set(), *[n[\"A1\"] for n in nB0]) | set.union(set(), *[n[\"A1\"] for n in nB1])\n",
    "newB0 = set.union(set(), *[n[\"B0\"] for n in nA0]) | set.union(set(), *[n[\"B0\"] for n in nA1])\n",
    "newB1 = set.union(set(), *[n[\"B1\"] for n in nA0]) | set.union(set(), *[n[\"B1\"] for n in nA1])\n",
    "A0 = A0 | newA0\n",
    "A1 = A1 | newA1\n",
    "B0 = B0 | newB0\n",
    "B1 = B1 | newB1\n",
    "print(f\"A0: {A0}\\nA1: {A1}\\nB0: {B0}\\nB1: {B1}\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "{-16, -8, 0, 8, 16}"
      ]
     },
     "execution_count": 37,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "{sum(v)//3 for v in A0}"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "{-12, -4, 4, 12}"
      ]
     },
     "execution_count": 38,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "{sum(v)//3 for v in A1}"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "{-9, -1, 7, 15}"
      ]
     },
     "execution_count": 34,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "{sum(v)//3 for v in B0}"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "{-13, -5, 3, 11}"
      ]
     },
     "execution_count": 35,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "{sum(v)//3 for v in B1}"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 163,
   "metadata": {},
   "outputs": [],
   "source": [
    "def to_mma(s):\n",
    "    return \"{\" + \", \".join([f\"{{{v[0]}, {v[1]}, {v[2]}}}\" for v in s]) + \"}\""
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 58,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "{(72, (-6, -6, 0)),\n",
       " (72, (-6, 0, -6)),\n",
       " (72, (0, -6, -6)),\n",
       " (72, (2, 2, 8)),\n",
       " (72, (2, 8, 2)),\n",
       " (72, (8, 2, 2)),\n",
       " (144, (-12, 0, 0)),\n",
       " (144, (-4, 8, 8)),\n",
       " (144, (0, -12, 0)),\n",
       " (144, (0, 0, -12)),\n",
       " (144, (8, -4, 8)),\n",
       " (144, (8, 8, -4)),\n",
       " (216, (-12, -6, 6)),\n",
       " (216, (-12, 6, -6)),\n",
       " (216, (-6, -12, 6)),\n",
       " (216, (-6, 6, -12)),\n",
       " (216, (-4, 2, 14)),\n",
       " (216, (-4, 14, 2)),\n",
       " (216, (2, -4, 14)),\n",
       " (216, (2, 14, -4)),\n",
       " (216, (6, -12, -6)),\n",
       " (216, (6, -6, -12)),\n",
       " (216, (14, -4, 2)),\n",
       " (216, (14, 2, -4)),\n",
       " (360, (-18, 0, 6)),\n",
       " (360, (-18, 6, 0)),\n",
       " (360, (-10, 8, 14)),\n",
       " (360, (-10, 14, 8)),\n",
       " (360, (0, -18, 6)),\n",
       " (360, (0, 6, -18)),\n",
       " (360, (6, -18, 0)),\n",
       " (360, (6, 0, -18)),\n",
       " (360, (8, -10, 14)),\n",
       " (360, (8, 14, -10)),\n",
       " (360, (14, -10, 8)),\n",
       " (360, (14, 8, -10)),\n",
       " (432, (-12, -12, 12)),\n",
       " (432, (-12, 12, -12)),\n",
       " (432, (-4, -4, 20)),\n",
       " (432, (-4, 20, -4)),\n",
       " (432, (12, -12, -12)),\n",
       " (432, (20, -4, -4)),\n",
       " (456, (-14, -14, -8)),\n",
       " (456, (-14, -8, -14)),\n",
       " (456, (-8, -14, -14)),\n",
       " (456, (10, 10, 16)),\n",
       " (456, (10, 16, 10)),\n",
       " (456, (16, 10, 10)),\n",
       " (504, (-18, -6, 12)),\n",
       " (504, (-18, 12, -6)),\n",
       " (504, (-10, 2, 20)),\n",
       " (504, (-10, 20, 2)),\n",
       " (504, (-6, -18, 12)),\n",
       " (504, (-6, 12, -18)),\n",
       " (504, (2, -10, 20)),\n",
       " (504, (2, 20, -10)),\n",
       " (504, (12, -18, -6)),\n",
       " (504, (12, -6, -18)),\n",
       " (504, (20, -10, 2)),\n",
       " (504, (20, 2, -10)),\n",
       " (528, (-20, -8, -8)),\n",
       " (528, (-8, -20, -8)),\n",
       " (528, (-8, -8, -20)),\n",
       " (528, (4, 16, 16)),\n",
       " (528, (16, 4, 16)),\n",
       " (528, (16, 16, 4)),\n",
       " (600, (-20, -14, -2)),\n",
       " (600, (-20, -2, -14)),\n",
       " (600, (-14, -20, -2)),\n",
       " (600, (-14, -2, -20)),\n",
       " (600, (-2, -20, -14)),\n",
       " (600, (-2, -14, -20)),\n",
       " (600, (4, 10, 22)),\n",
       " (600, (4, 22, 10)),\n",
       " (600, (10, 4, 22)),\n",
       " (600, (10, 22, 4)),\n",
       " (600, (22, 4, 10)),\n",
       " (600, (22, 10, 4)),\n",
       " (648, (-24, 6, 6)),\n",
       " (648, (-16, 14, 14)),\n",
       " (648, (6, -24, 6)),\n",
       " (648, (6, 6, -24)),\n",
       " (648, (14, -16, 14)),\n",
       " (648, (14, 14, -16)),\n",
       " (720, (-24, 0, 12)),\n",
       " (720, (-24, 12, 0)),\n",
       " (720, (-16, 8, 20)),\n",
       " (720, (-16, 20, 8)),\n",
       " (720, (0, -24, 12)),\n",
       " (720, (0, 12, -24)),\n",
       " (720, (8, -16, 20)),\n",
       " (720, (8, 20, -16)),\n",
       " (720, (12, -24, 0)),\n",
       " (720, (12, 0, -24)),\n",
       " (720, (20, -16, 8)),\n",
       " (720, (20, 8, -16)),\n",
       " (744, (-26, -8, -2)),\n",
       " (744, (-26, -2, -8)),\n",
       " (744, (-8, -26, -2)),\n",
       " (744, (-8, -2, -26)),\n",
       " (744, (-2, -26, -8)),\n",
       " (744, (-2, -8, -26)),\n",
       " (744, (-2, 16, 22)),\n",
       " (744, (-2, 22, 16)),\n",
       " (744, (16, -2, 22)),\n",
       " (744, (16, 22, -2)),\n",
       " (744, (22, -2, 16)),\n",
       " (744, (22, 16, -2)),\n",
       " (792, (-18, -12, 18)),\n",
       " (792, (-18, 18, -12)),\n",
       " (792, (-12, -18, 18)),\n",
       " (792, (-12, 18, -18)),\n",
       " (792, (-10, -4, 26)),\n",
       " (792, (-10, 26, -4)),\n",
       " (792, (-4, -10, 26)),\n",
       " (792, (-4, 26, -10)),\n",
       " (792, (18, -18, -12)),\n",
       " (792, (18, -12, -18)),\n",
       " (792, (26, -10, -4)),\n",
       " (792, (26, -4, -10)),\n",
       " (816, (-20, -20, 4)),\n",
       " (816, (-20, 4, -20)),\n",
       " (816, (4, -20, -20)),\n",
       " (816, (4, 4, 28)),\n",
       " (816, (4, 28, 4)),\n",
       " (816, (28, 4, 4)),\n",
       " (888, (-26, -14, 4)),\n",
       " (888, (-26, 4, -14)),\n",
       " (888, (-14, -26, 4)),\n",
       " (888, (-14, 4, -26)),\n",
       " (888, (-2, 10, 28)),\n",
       " (888, (-2, 28, 10)),\n",
       " (888, (4, -26, -14)),\n",
       " (888, (4, -14, -26)),\n",
       " (888, (10, -2, 28)),\n",
       " (888, (10, 28, -2)),\n",
       " (888, (28, -2, 10)),\n",
       " (888, (28, 10, -2)),\n",
       " (936, (-24, -6, 18)),\n",
       " (936, (-24, 18, -6)),\n",
       " (936, (-16, 2, 26)),\n",
       " (936, (-16, 26, 2)),\n",
       " (936, (-6, -24, 18)),\n",
       " (936, (-6, 18, -24)),\n",
       " (936, (2, -16, 26)),\n",
       " (936, (2, 26, -16)),\n",
       " (936, (18, -24, -6)),\n",
       " (936, (18, -6, -24)),\n",
       " (936, (26, -16, 2)),\n",
       " (936, (26, 2, -16))}"
      ]
     },
     "execution_count": 58,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "{(v[0]**2+v[1]**2+v[2]**2,(v[0], v[1], v[2])) for v in A1}"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 40,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "{(72, (-6, -6, 0)),\n",
       " (72, (-6, 0, -6)),\n",
       " (72, (0, -6, -6)),\n",
       " (72, (2, 2, 8)),\n",
       " (72, (2, 8, 2)),\n",
       " (72, (8, 2, 2)),\n",
       " (144, (-12, 0, 0)),\n",
       " (144, (-4, 8, 8)),\n",
       " (144, (0, -12, 0)),\n",
       " (144, (0, 0, -12)),\n",
       " (144, (8, -4, 8)),\n",
       " (144, (8, 8, -4)),\n",
       " (216, (-12, -6, 6)),\n",
       " (216, (-12, 6, -6)),\n",
       " (216, (-6, -12, 6)),\n",
       " (216, (-6, 6, -12)),\n",
       " (216, (-4, 2, 14)),\n",
       " (216, (-4, 14, 2)),\n",
       " (216, (2, -4, 14)),\n",
       " (216, (2, 14, -4)),\n",
       " (216, (6, -12, -6)),\n",
       " (216, (6, -6, -12)),\n",
       " (216, (14, -4, 2)),\n",
       " (216, (14, 2, -4)),\n",
       " (360, (-18, 0, 6)),\n",
       " (360, (-18, 6, 0)),\n",
       " (360, (-10, 8, 14)),\n",
       " (360, (-10, 14, 8)),\n",
       " (360, (0, -18, 6)),\n",
       " (360, (0, 6, -18)),\n",
       " (360, (6, -18, 0)),\n",
       " (360, (6, 0, -18)),\n",
       " (360, (8, -10, 14)),\n",
       " (360, (8, 14, -10)),\n",
       " (360, (14, -10, 8)),\n",
       " (360, (14, 8, -10)),\n",
       " (432, (-12, -12, 12)),\n",
       " (432, (-12, 12, -12)),\n",
       " (432, (-4, -4, 20)),\n",
       " (432, (-4, 20, -4)),\n",
       " (432, (12, -12, -12)),\n",
       " (432, (20, -4, -4)),\n",
       " (456, (-14, -14, -8)),\n",
       " (456, (-14, -8, -14)),\n",
       " (456, (-8, -14, -14)),\n",
       " (456, (10, 10, 16)),\n",
       " (456, (10, 16, 10)),\n",
       " (456, (16, 10, 10)),\n",
       " (504, (-18, -6, 12)),\n",
       " (504, (-18, 12, -6)),\n",
       " (504, (-10, 2, 20)),\n",
       " (504, (-10, 20, 2)),\n",
       " (504, (-6, -18, 12)),\n",
       " (504, (-6, 12, -18)),\n",
       " (504, (2, -10, 20)),\n",
       " (504, (2, 20, -10)),\n",
       " (504, (12, -18, -6)),\n",
       " (504, (12, -6, -18)),\n",
       " (504, (20, -10, 2)),\n",
       " (504, (20, 2, -10)),\n",
       " (528, (-20, -8, -8)),\n",
       " (528, (-8, -20, -8)),\n",
       " (528, (-8, -8, -20)),\n",
       " (528, (4, 16, 16)),\n",
       " (528, (16, 4, 16)),\n",
       " (528, (16, 16, 4)),\n",
       " (600, (-20, -14, -2)),\n",
       " (600, (-20, -2, -14)),\n",
       " (600, (-14, -20, -2)),\n",
       " (600, (-14, -2, -20)),\n",
       " (600, (-2, -20, -14)),\n",
       " (600, (-2, -14, -20)),\n",
       " (600, (4, 10, 22)),\n",
       " (600, (4, 22, 10)),\n",
       " (600, (10, 4, 22)),\n",
       " (600, (10, 22, 4)),\n",
       " (600, (22, 4, 10)),\n",
       " (600, (22, 10, 4)),\n",
       " (648, (-24, 6, 6)),\n",
       " (648, (-16, 14, 14)),\n",
       " (648, (6, -24, 6)),\n",
       " (648, (6, 6, -24)),\n",
       " (648, (14, -16, 14)),\n",
       " (648, (14, 14, -16)),\n",
       " (720, (-24, 0, 12)),\n",
       " (720, (-24, 12, 0)),\n",
       " (720, (-16, 8, 20)),\n",
       " (720, (-16, 20, 8)),\n",
       " (720, (0, -24, 12)),\n",
       " (720, (0, 12, -24)),\n",
       " (720, (8, -16, 20)),\n",
       " (720, (8, 20, -16)),\n",
       " (720, (12, -24, 0)),\n",
       " (720, (12, 0, -24)),\n",
       " (720, (20, -16, 8)),\n",
       " (720, (20, 8, -16)),\n",
       " (744, (-26, -8, -2)),\n",
       " (744, (-26, -2, -8)),\n",
       " (744, (-8, -26, -2)),\n",
       " (744, (-8, -2, -26)),\n",
       " (744, (-2, -26, -8)),\n",
       " (744, (-2, -8, -26)),\n",
       " (744, (-2, 16, 22)),\n",
       " (744, (-2, 22, 16)),\n",
       " (744, (16, -2, 22)),\n",
       " (744, (16, 22, -2)),\n",
       " (744, (22, -2, 16)),\n",
       " (744, (22, 16, -2)),\n",
       " (792, (-18, -12, 18)),\n",
       " (792, (-18, 18, -12)),\n",
       " (792, (-12, -18, 18)),\n",
       " (792, (-12, 18, -18)),\n",
       " (792, (-10, -4, 26)),\n",
       " (792, (-10, 26, -4)),\n",
       " (792, (-4, -10, 26)),\n",
       " (792, (-4, 26, -10)),\n",
       " (792, (18, -18, -12)),\n",
       " (792, (18, -12, -18)),\n",
       " (792, (26, -10, -4)),\n",
       " (792, (26, -4, -10)),\n",
       " (816, (-20, -20, 4)),\n",
       " (816, (-20, 4, -20)),\n",
       " (816, (4, -20, -20)),\n",
       " (816, (4, 4, 28)),\n",
       " (816, (4, 28, 4)),\n",
       " (816, (28, 4, 4)),\n",
       " (888, (-26, -14, 4)),\n",
       " (888, (-26, 4, -14)),\n",
       " (888, (-14, -26, 4)),\n",
       " (888, (-14, 4, -26)),\n",
       " (888, (-2, 10, 28)),\n",
       " (888, (-2, 28, 10)),\n",
       " (888, (4, -26, -14)),\n",
       " (888, (4, -14, -26)),\n",
       " (888, (10, -2, 28)),\n",
       " (888, (10, 28, -2)),\n",
       " (888, (28, -2, 10)),\n",
       " (888, (28, 10, -2)),\n",
       " (936, (-24, -6, 18)),\n",
       " (936, (-24, 18, -6)),\n",
       " (936, (-16, 2, 26)),\n",
       " (936, (-16, 26, 2)),\n",
       " (936, (-6, -24, 18)),\n",
       " (936, (-6, 18, -24)),\n",
       " (936, (2, -16, 26)),\n",
       " (936, (2, 26, -16)),\n",
       " (936, (18, -24, -6)),\n",
       " (936, (18, -6, -24)),\n",
       " (936, (26, -16, 2)),\n",
       " (936, (26, 2, -16))}"
      ]
     },
     "execution_count": 40,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "{(v[0]**2+v[1]**2+v[2]**2,v) for v in A1}"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "2"
      ]
     },
     "execution_count": 1,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "1+1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "interpreter": {
   "hash": "7a360a95d56bf1be2ed957b4970aa89f6b1c3bbe90d8df6643aa18a01e23f4ef"
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   "display_name": "Python 3.10.1 64-bit ('base': conda)",
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    "name": "ipython",
    "version": 3
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   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
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