fixed things hopefully
This commit is contained in:
William Ball
parent
d26a1874a9
commit
a7a3191240
2 changed files with 70 additions and 178 deletions
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@ -16,7 +16,7 @@
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},
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"cell_type": "code",
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"execution_count": 49,
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"execution_count": 1,
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"metadata": {},
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"outputs": [],
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@ -30,7 +30,7 @@
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{
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"cell_type": "code",
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"execution_count": 50,
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"execution_count": 2,
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"metadata": {},
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"outputs": [
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{
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@ -42,7 +42,7 @@
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"[ 0 0 0 1]"
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]
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},
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"execution_count": 50,
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"execution_count": 2,
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"metadata": {},
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"output_type": "execute_result"
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}
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@ -62,7 +62,7 @@
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},
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{
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"cell_type": "code",
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"execution_count": 51,
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"execution_count": 3,
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"metadata": {},
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"outputs": [],
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"source": [
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@ -99,7 +99,7 @@
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},
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{
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"cell_type": "code",
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"execution_count": 52,
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"execution_count": 4,
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"metadata": {
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"scrolled": true
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},
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@ -113,7 +113,7 @@
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"[-1 -1 -1 -1]"
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]
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},
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"execution_count": 52,
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"execution_count": 4,
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"metadata": {},
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"output_type": "execute_result"
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}
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@ -125,7 +125,7 @@
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},
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{
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"cell_type": "code",
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"execution_count": 53,
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"execution_count": 5,
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"metadata": {
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"scrolled": true
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},
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@ -139,7 +139,7 @@
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"[-1/2 -1/2 -1/2 1/2]"
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]
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},
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"execution_count": 53,
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"execution_count": 5,
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"metadata": {},
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"output_type": "execute_result"
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}
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@ -150,7 +150,7 @@
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},
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{
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"cell_type": "code",
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"execution_count": 54,
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"execution_count": 6,
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"metadata": {},
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"outputs": [],
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"source": [
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@ -182,7 +182,7 @@
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},
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{
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"cell_type": "code",
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"execution_count": 55,
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"execution_count": 7,
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"metadata": {},
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"outputs": [],
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"source": [
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@ -191,7 +191,7 @@
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},
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{
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"cell_type": "code",
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"execution_count": 56,
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"execution_count": 8,
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"metadata": {},
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"outputs": [
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{
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@ -203,7 +203,7 @@
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"[ 0 0 0 1]"
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]
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},
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"execution_count": 56,
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"execution_count": 8,
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"metadata": {},
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"output_type": "execute_result"
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}
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@ -214,7 +214,7 @@
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},
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{
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"cell_type": "code",
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"execution_count": 57,
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"execution_count": 9,
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"metadata": {},
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"outputs": [
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{
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@ -223,7 +223,7 @@
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"[b_avg == b1 + b2 + b3 - sqrt(2*b1*b2 + 2*(b1 + b2)*b3), b_avg == b1 + b2 + b3 + sqrt(2*b1*b2 + 2*(b1 + b2)*b3)]"
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]
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},
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"execution_count": 57,
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"execution_count": 9,
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"metadata": {},
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"output_type": "execute_result"
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}
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@ -235,7 +235,7 @@
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},
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{
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"cell_type": "code",
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"execution_count": 58,
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"execution_count": 10,
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"metadata": {},
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"outputs": [
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{
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@ -244,7 +244,7 @@
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"b1^2 + b2^2 + b3^2 - 2*b1*b_avg - 2*b2*b_avg - 2*b3*b_avg + b_avg^2"
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]
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},
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"execution_count": 58,
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"execution_count": 10,
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"metadata": {},
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"output_type": "execute_result"
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}
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@ -266,7 +266,7 @@
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},
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{
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"cell_type": "code",
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"execution_count": 59,
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"execution_count": 11,
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"metadata": {},
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"outputs": [
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{
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@ -275,7 +275,7 @@
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"1/2*b1*h11 - 1/2*b_avg*h11 - 1/2*b1*h1_avg - 1/2*b2*h1_avg - 1/2*b3*h1_avg + 1/2*b_avg*h1_avg + 1/2*b2*h21 - 1/2*b_avg*h21 + 1/2*b3*h31 - 1/2*b_avg*h31"
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]
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},
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"execution_count": 59,
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"execution_count": 11,
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"metadata": {},
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"output_type": "execute_result"
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}
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@ -290,12 +290,12 @@
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"source": [
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"# Calculations to find the generators for the \"Apollonian\" Group for the octahedral packing\n",
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"\n",
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"We can try the reflection group, since that is one way to derive the generators for the Apollonian group and see if it works. My worry is that it won't since the coordinate system is so different."
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"We can try the Weyl group, since that is one way to derive the generators for the Apollonian group and see if it works. My worry is that it won't since the coordinate system is so different."
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]
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},
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{
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"cell_type": "code",
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"execution_count": 60,
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"execution_count": 12,
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"metadata": {},
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"outputs": [],
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"source": [
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@ -331,7 +331,7 @@
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},
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{
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"cell_type": "code",
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"execution_count": 76,
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"execution_count": 13,
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"metadata": {},
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"outputs": [],
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"source": [
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@ -345,7 +345,7 @@
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},
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{
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"cell_type": "code",
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"execution_count": 85,
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"execution_count": 14,
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"metadata": {},
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"outputs": [],
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"source": [
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@ -373,70 +373,16 @@
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" return drawing"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 95,
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"metadata": {},
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"outputs": [],
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"source": [
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"def draw_circles_from_coords(coords, color=(0,0,1)):\n",
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" drawing = None\n",
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" for row in coords:\n",
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" b = row[1]\n",
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" if not b == 0:\n",
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" x = row[2] / b\n",
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" y = row[3] / b\n",
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" if drawing is not None:\n",
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" drawing += circle((x, y), 1 / b, rgbcolor=color)\n",
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" else:\n",
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" drawing = circle((x, y), 1 / b, rgbcolor=color)\n",
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" return drawing"
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"metadata": {},
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"outputs": [],
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"source": [
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"root = matrix([\n",
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" [2, 0, 0, 1],\n",
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" [2, 0, 0, -1],\n",
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" [-1, 1, 0, 0],\n",
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" [3, 1, 2, 0]\n",
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"])\n",
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"\n",
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"S4 = matrix([\n",
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" [1, 0, 0, 0],\n",
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" [0, 1, 0, 0],\n",
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" [0, 0, 1, 0],\n",
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" [2, 2, 2, -1],\n",
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"])\n",
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"\n",
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"drawing = draw_circles_from_coords(root)\n",
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"branches = [root]\n",
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"for i in range(10):\n",
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" new_branches = []\n",
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" for tree in branches:\n",
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" drawing += draw_circles_from_coords(tree)\n",
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" new_branches.append(S1 * tree)\n",
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" new_branches.append(S2 * tree)\n",
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" new_branches.append(S3 * tree)\n",
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" new_branches.append(S4 * tree)\n",
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" branches = new_branches\n",
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"drawing"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"Hmm. Doesn't look like it worked. And my crude code to draw the circles doesn't seem to be the problem either."
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"Hmm. Doesn't look like it worked. And my crude code to draw the circles doesn't seem to be the problem either. I calculated a few of these circles by hand using the matrices and it comes out exactly where it's drawn here, so my code isn't the problem. Unless, of course, I computed the generators of the Weyl group wrong."
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]
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},
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{
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"cell_type": "code",
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"execution_count": 93,
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"execution_count": 15,
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"metadata": {},
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"outputs": [
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{
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"Graphics object consisting of 8 graphics primitives"
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]
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},
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"execution_count": 93,
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"execution_count": 15,
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"metadata": {},
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"output_type": "execute_result"
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}
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@ -466,7 +412,7 @@
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},
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{
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"cell_type": "code",
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"execution_count": 63,
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"execution_count": 16,
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"metadata": {},
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"outputs": [],
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"source": [
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@ -490,7 +436,7 @@
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},
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{
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"cell_type": "code",
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"execution_count": 64,
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"execution_count": 17,
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"metadata": {},
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"outputs": [
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{
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@ -555,7 +501,7 @@
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},
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{
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"cell_type": "code",
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"execution_count": 65,
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"execution_count": 18,
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"metadata": {},
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"outputs": [],
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"source": [
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},
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{
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"cell_type": "code",
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"execution_count": 66,
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"execution_count": 19,
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"metadata": {},
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"outputs": [
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{
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"[ 3/4 -3/2 3/4 -1 -5/4 1/2 -5/4 1]"
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]
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},
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"execution_count": 66,
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"execution_count": 19,
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"metadata": {},
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"output_type": "execute_result"
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}
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},
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{
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"cell_type": "code",
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"execution_count": 67,
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"execution_count": 20,
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"metadata": {},
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"outputs": [],
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"source": [
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},
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{
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"cell_type": "code",
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"execution_count": 68,
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"execution_count": 21,
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"metadata": {
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"scrolled": true
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},
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"[-5/4 1/2 -5/4 1]"
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]
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},
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"execution_count": 68,
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"execution_count": 21,
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"metadata": {},
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"output_type": "execute_result"
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}
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},
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{
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"cell_type": "code",
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"execution_count": 69,
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"execution_count": 22,
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"metadata": {},
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"outputs": [
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{
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@ -653,7 +599,7 @@
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"97*b1^2 - 304*b1*b2 + 328*b2^2 - 290*b1*b3 - 304*b2*b3 + 97*b3^2 + 32*b1*b_avg - 1088*b2*b_avg + 32*b3*b_avg + 1280*b_avg^2"
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]
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},
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"execution_count": 69,
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"execution_count": 22,
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"metadata": {},
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"output_type": "execute_result"
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}
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},
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{
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"cell_type": "code",
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"execution_count": 49,
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"execution_count": 1,
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"metadata": {},
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"outputs": [],
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"source": [
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{
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"cell_type": "code",
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"execution_count": 50,
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"execution_count": 2,
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"outputs": [
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{
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]
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},
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"execution_count": 50,
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"output_type": "execute_result"
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}
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},
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{
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"execution_count": 51,
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"metadata": {},
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"outputs": [],
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"source": [
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{
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"cell_type": "code",
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"execution_count": 52,
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"metadata": {
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"scrolled": true
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},
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"[-1 -1 -1 -1]"
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]
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},
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"execution_count": 52,
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"output_type": "execute_result"
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"execution_count": 53,
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"scrolled": true
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"output_type": "execute_result"
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"execution_count": 55,
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"outputs": [],
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{
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"cell_type": "code",
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"execution_count": 56,
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"metadata": {},
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"outputs": [
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{
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"[ 0 0 0 1]"
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]
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},
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"execution_count": 56,
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"execution_count": 8,
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"metadata": {},
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"output_type": "execute_result"
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},
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{
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"cell_type": "code",
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"execution_count": 57,
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"execution_count": 9,
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"metadata": {},
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"outputs": [
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{
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"[b_avg == b1 + b2 + b3 - sqrt(2*b1*b2 + 2*(b1 + b2)*b3), b_avg == b1 + b2 + b3 + sqrt(2*b1*b2 + 2*(b1 + b2)*b3)]"
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]
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},
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"execution_count": 57,
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"output_type": "execute_result"
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}
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},
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{
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"cell_type": "code",
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"execution_count": 58,
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"execution_count": 10,
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"metadata": {},
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"outputs": [
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{
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|
@ -244,7 +244,7 @@
|
|||
"b1^2 + b2^2 + b3^2 - 2*b1*b_avg - 2*b2*b_avg - 2*b3*b_avg + b_avg^2"
|
||||
]
|
||||
},
|
||||
"execution_count": 58,
|
||||
"execution_count": 10,
|
||||
"metadata": {},
|
||||
"output_type": "execute_result"
|
||||
}
|
||||
|
|
@ -266,7 +266,7 @@
|
|||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 59,
|
||||
"execution_count": 11,
|
||||
"metadata": {},
|
||||
"outputs": [
|
||||
{
|
||||
|
|
@ -275,7 +275,7 @@
|
|||
"1/2*b1*h11 - 1/2*b_avg*h11 - 1/2*b1*h1_avg - 1/2*b2*h1_avg - 1/2*b3*h1_avg + 1/2*b_avg*h1_avg + 1/2*b2*h21 - 1/2*b_avg*h21 + 1/2*b3*h31 - 1/2*b_avg*h31"
|
||||
]
|
||||
},
|
||||
"execution_count": 59,
|
||||
"execution_count": 11,
|
||||
"metadata": {},
|
||||
"output_type": "execute_result"
|
||||
}
|
||||
|
|
@ -290,12 +290,12 @@
|
|||
"source": [
|
||||
"# Calculations to find the generators for the \"Apollonian\" Group for the octahedral packing\n",
|
||||
"\n",
|
||||
"We can try the reflection group, since that is one way to derive the generators for the Apollonian group and see if it works. My worry is that it won't since the coordinate system is so different."
|
||||
"We can try the Weyl group, since that is one way to derive the generators for the Apollonian group and see if it works. My worry is that it won't since the coordinate system is so different."
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 60,
|
||||
"execution_count": 12,
|
||||
"metadata": {},
|
||||
"outputs": [],
|
||||
"source": [
|
||||
|
|
@ -331,7 +331,7 @@
|
|||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 76,
|
||||
"execution_count": 13,
|
||||
"metadata": {},
|
||||
"outputs": [],
|
||||
"source": [
|
||||
|
|
@ -345,7 +345,7 @@
|
|||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 85,
|
||||
"execution_count": 14,
|
||||
"metadata": {},
|
||||
"outputs": [],
|
||||
"source": [
|
||||
|
|
@ -373,70 +373,16 @@
|
|||
" return drawing"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 95,
|
||||
"metadata": {},
|
||||
"outputs": [],
|
||||
"source": [
|
||||
"def draw_circles_from_coords(coords, color=(0,0,1)):\n",
|
||||
" drawing = None\n",
|
||||
" for row in coords:\n",
|
||||
" b = row[1]\n",
|
||||
" if not b == 0:\n",
|
||||
" x = row[2] / b\n",
|
||||
" y = row[3] / b\n",
|
||||
" if drawing is not None:\n",
|
||||
" drawing += circle((x, y), 1 / b, rgbcolor=color)\n",
|
||||
" else:\n",
|
||||
" drawing = circle((x, y), 1 / b, rgbcolor=color)\n",
|
||||
" return drawing"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": null,
|
||||
"metadata": {},
|
||||
"outputs": [],
|
||||
"source": [
|
||||
"root = matrix([\n",
|
||||
" [2, 0, 0, 1],\n",
|
||||
" [2, 0, 0, -1],\n",
|
||||
" [-1, 1, 0, 0],\n",
|
||||
" [3, 1, 2, 0]\n",
|
||||
"])\n",
|
||||
"\n",
|
||||
"S4 = matrix([\n",
|
||||
" [1, 0, 0, 0],\n",
|
||||
" [0, 1, 0, 0],\n",
|
||||
" [0, 0, 1, 0],\n",
|
||||
" [2, 2, 2, -1],\n",
|
||||
"])\n",
|
||||
"\n",
|
||||
"drawing = draw_circles_from_coords(root)\n",
|
||||
"branches = [root]\n",
|
||||
"for i in range(10):\n",
|
||||
" new_branches = []\n",
|
||||
" for tree in branches:\n",
|
||||
" drawing += draw_circles_from_coords(tree)\n",
|
||||
" new_branches.append(S1 * tree)\n",
|
||||
" new_branches.append(S2 * tree)\n",
|
||||
" new_branches.append(S3 * tree)\n",
|
||||
" new_branches.append(S4 * tree)\n",
|
||||
" branches = new_branches\n",
|
||||
"drawing"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"Hmm. Doesn't look like it worked. And my crude code to draw the circles doesn't seem to be the problem either."
|
||||
"Hmm. Doesn't look like it worked. And my crude code to draw the circles doesn't seem to be the problem either. I calculated a few of these circles by hand using the matrices and it comes out exactly where it's drawn here, so my code isn't the problem. Unless, of course, I computed the generators of the Weyl group wrong."
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 93,
|
||||
"execution_count": 15,
|
||||
"metadata": {},
|
||||
"outputs": [
|
||||
{
|
||||
|
|
@ -446,7 +392,7 @@
|
|||
"Graphics object consisting of 8 graphics primitives"
|
||||
]
|
||||
},
|
||||
"execution_count": 93,
|
||||
"execution_count": 15,
|
||||
"metadata": {},
|
||||
"output_type": "execute_result"
|
||||
}
|
||||
|
|
@ -466,7 +412,7 @@
|
|||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 63,
|
||||
"execution_count": 16,
|
||||
"metadata": {},
|
||||
"outputs": [],
|
||||
"source": [
|
||||
|
|
@ -490,7 +436,7 @@
|
|||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 64,
|
||||
"execution_count": 17,
|
||||
"metadata": {},
|
||||
"outputs": [
|
||||
{
|
||||
|
|
@ -555,7 +501,7 @@
|
|||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 65,
|
||||
"execution_count": 18,
|
||||
"metadata": {},
|
||||
"outputs": [],
|
||||
"source": [
|
||||
|
|
@ -565,7 +511,7 @@
|
|||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 66,
|
||||
"execution_count": 19,
|
||||
"metadata": {},
|
||||
"outputs": [
|
||||
{
|
||||
|
|
@ -581,7 +527,7 @@
|
|||
"[ 3/4 -3/2 3/4 -1 -5/4 1/2 -5/4 1]"
|
||||
]
|
||||
},
|
||||
"execution_count": 66,
|
||||
"execution_count": 19,
|
||||
"metadata": {},
|
||||
"output_type": "execute_result"
|
||||
}
|
||||
|
|
@ -604,7 +550,7 @@
|
|||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 67,
|
||||
"execution_count": 20,
|
||||
"metadata": {},
|
||||
"outputs": [],
|
||||
"source": [
|
||||
|
|
@ -618,7 +564,7 @@
|
|||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 68,
|
||||
"execution_count": 21,
|
||||
"metadata": {
|
||||
"scrolled": true
|
||||
},
|
||||
|
|
@ -632,7 +578,7 @@
|
|||
"[-5/4 1/2 -5/4 1]"
|
||||
]
|
||||
},
|
||||
"execution_count": 68,
|
||||
"execution_count": 21,
|
||||
"metadata": {},
|
||||
"output_type": "execute_result"
|
||||
}
|
||||
|
|
@ -644,7 +590,7 @@
|
|||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 69,
|
||||
"execution_count": 22,
|
||||
"metadata": {},
|
||||
"outputs": [
|
||||
{
|
||||
|
|
@ -653,7 +599,7 @@
|
|||
"97*b1^2 - 304*b1*b2 + 328*b2^2 - 290*b1*b3 - 304*b2*b3 + 97*b3^2 + 32*b1*b_avg - 1088*b2*b_avg + 32*b3*b_avg + 1280*b_avg^2"
|
||||
]
|
||||
},
|
||||
"execution_count": 69,
|
||||
"execution_count": 22,
|
||||
"metadata": {},
|
||||
"output_type": "execute_result"
|
||||
}
|
||||
|
|
|
|||
Loading…
Reference in a new issue