Mandala-Like Figures from Joining Points
Author
Stephen Wolfram
Title
Mandala-Like Figures from Joining Points
Description
Here are 5 points arranged around a circle:
Category
Educational Materials
Keywords
URL
http://www.notebookarchive.org/2019-08-98zxi0w/
DOI
https://notebookarchive.org/2019-08-98zxi0w
Date Added
2019-08-20
Date Last Modified
2019-08-20
File Size
1.23 megabytes
Supplements
Rights
Redistribution rights reserved
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Mandala-Like Figures from Joining Points
Mandala-Like Figures from Joining Points
Here are 5 points arranged around a circle:
In[]:=
Graphics[Point[CirclePoints[5]]]
Out[]=
This joins the points in pairs:
In[]:=
Graphics[Line[Tuples[CirclePoints[5],2]]]
Out[]=
Here’s how Tuples works, picking all possible pairs out of {a,b,c}:
In[]:=
Tuples[{a,b,c},2]
Out[]=
{{a,a},{a,b},{a,c},{b,a},{b,b},{b,c},{c,a},{c,b},{c,c}}
Here’s what it does picking from {a,b,c,d}:
In[]:=
Tuples[{a,b,c,d},2]
Out[]=
{{a,a},{a,b},{a,c},{a,d},{b,a},{b,b},{b,c},{b,d},{c,a},{c,b},{c,c},{c,d},{d,a},{d,b},{d,c},{d,d}}
Join all possible pairs of 4 points:
In[]:=
Graphics[Line[Tuples[CirclePoints[4],2]]]
Out[]=
6 points:
In[]:=
Graphics[Line[Tuples[CirclePoints[6],2]]]
Out[]=
7 points:
In[]:=
Graphics[Line[Tuples[CirclePoints[7],2]]]
Out[]=
11 points:
In[]:=
Graphics[Line[Tuples[CirclePoints[11],2]]]
Out[]=
50 points:
In[]:=
Graphics[Line[Tuples[CirclePoints[50],2]]]
Out[]=
With an odd number of points---like 49---the center is white:
In[]:=
Graphics[Line[Tuples[CirclePoints[49],2]]]
Out[]=
The 3D Case
The 3D Case
SpherePoints distributes points as uniformly as possible on a sphere:
In[]:=
Graphics3D[Point[SpherePoints[20]]]
Out[]=
Join pairs of points in 3D:
In[]:=
Graphics3D[Line[Tuples[SpherePoints[20],2]]]
Out[]=
SpherePoints[12] makes a regular figure with 12 vertices:
In[]:=
Graphics3D[Line[Tuples[SpherePoints[12],2]]]
Out[]=
The figure is an icosahedron:
In[]:=
PolyhedronData["Icosahedron"]
Out[]=
Here is a list of the Platonic solids:
In[]:=
PolyhedronData["Platonic"]
Out[]=
{Tetrahedron,Cube,Octahedron,Dodecahedron,Icosahedron}
Here are pictures of them:
In[]:=
PolyhedronData/@%
Out[]=
,
,
,
,
There are other polyhedra that aren’t Platonic:
In[]:=
PolyhedronData["RhombicHexecontahedron"]
Out[]=
This joins pairs of 25 points in 3D:
In[]:=
Graphics3D[Line[Tuples[SpherePoints[25],2]]]
Out[]=
There is no Platonic solid with 25 points, but SpherePoints distributes the points as uniformly as possible in 3D.
The corresponding result for 20 points:
In[]:=
Graphics3D[Line[Tuples[SpherePoints[20],2]]]
Out[]=
Reduced Versions
Reduced Versions
Make an interactive version that includes only lines above a specified length:
In[]:=
Manipulate[Graphics[Line[Select[Tuples[CirclePoints[5],2],Norm[#]>r&]]],{r,1,2}]
Out[]=
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In[]:=
Manipulate[Graphics[Line[Select[Tuples[CirclePoints[10],2],Norm[#]>r&]]],{r,1,2}]
Out[]=
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In[]:=
Manipulate[Graphics[Line[Select[Tuples[N[CirclePoints[21]],2],Norm[#]>r&]]],{r,1,2}]
Out[]=
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Cite this as: Stephen Wolfram, "Mandala-Like Figures from Joining Points" from the Notebook Archive (2019), https://notebookarchive.org/2019-08-98zxi0w
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