Platonic solids#

EXAMPLES: The five platonic solids in a row:

sage: G = tetrahedron((0,-3.5,0), color='blue') + cube((0,-2,0),color=(.25,0,.5))
sage: G += octahedron(color='red') + dodecahedron((0,2,0), color='orange')
sage: G += icosahedron(center=(0,4,0), color='yellow')
sage: G.show(aspect_ratio=[1,1,1])

>>> from sage.all import *
>>> G = tetrahedron((Integer(0),-RealNumber('3.5'),Integer(0)), color='blue') + cube((Integer(0),-Integer(2),Integer(0)),color=(RealNumber('.25'),Integer(0),RealNumber('.5')))
>>> G += octahedron(color='red') + dodecahedron((Integer(0),Integer(2),Integer(0)), color='orange')
>>> G += icosahedron(center=(Integer(0),Integer(4),Integer(0)), color='yellow')
>>> G.show(aspect_ratio=[Integer(1),Integer(1),Integer(1)])
../../../_images/platonic-1.svg

All the platonic solids in the same place:

sage: G = tetrahedron(color='blue',opacity=0.7)
sage: G += cube(color=(.25,0,.5), opacity=0.7)
sage: G += octahedron(color='red', opacity=0.7)
sage: G += dodecahedron(color='orange', opacity=0.7) + icosahedron(opacity=0.7)
sage: G.show(aspect_ratio=[1,1,1])

>>> from sage.all import *
>>> G = tetrahedron(color='blue',opacity=RealNumber('0.7'))
>>> G += cube(color=(RealNumber('.25'),Integer(0),RealNumber('.5')), opacity=RealNumber('0.7'))
>>> G += octahedron(color='red', opacity=RealNumber('0.7'))
>>> G += dodecahedron(color='orange', opacity=RealNumber('0.7')) + icosahedron(opacity=RealNumber('0.7'))
>>> G.show(aspect_ratio=[Integer(1),Integer(1),Integer(1)])
../../../_images/platonic-2.svg

Display nice faces only:

sage: icosahedron().stickers(['red','blue'], .075, .1)
Graphics3d Object

>>> from sage.all import *
>>> icosahedron().stickers(['red','blue'], RealNumber('.075'), RealNumber('.1'))
Graphics3d Object
../../../_images/platonic-3.svg

AUTHORS:

  • Robert Bradshaw (2007, 2008): initial version

  • William Stein

sage.plot.plot3d.platonic.cube(center=(0, 0, 0), size=1, color=None, frame_thickness=0, frame_color=None, **kwds)[source]#

A 3D cube centered at the origin with default side lengths 1.

INPUT:

  • center – (default: (0,0,0))

  • size – (default: 1) the side lengths of the cube

  • color – a string that describes a color; this can also be a list of 3-tuples or strings length 6 or 3, in which case the faces (and oppositive faces) are colored.

  • frame_thickness – (default: 0) if positive, then thickness of the frame

  • frame_color – (default: None) if given, gives the color of the frame

  • opacity – (default: 1) if less than 1 then it’s transparent

EXAMPLES:

A simple cube:

sage: cube()
Graphics3d Object

>>> from sage.all import *
>>> cube()
Graphics3d Object
../../../_images/platonic-4.svg

A red cube:

sage: cube(color="red")
Graphics3d Object

>>> from sage.all import *
>>> cube(color="red")
Graphics3d Object
../../../_images/platonic-5.svg

A transparent grey cube that contains a red cube:

sage: cube(opacity=0.8, color='grey') + cube(size=3/4)
Graphics3d Object

>>> from sage.all import *
>>> cube(opacity=RealNumber('0.8'), color='grey') + cube(size=Integer(3)/Integer(4))
Graphics3d Object
../../../_images/platonic-6.svg

A transparent colored cube:

sage: cube(color=['red', 'green', 'blue'], opacity=0.5)
Graphics3d Object

>>> from sage.all import *
>>> cube(color=['red', 'green', 'blue'], opacity=RealNumber('0.5'))
Graphics3d Object
../../../_images/platonic-7.svg

A bunch of random cubes:

sage: v = [(random(), random(), random()) for _ in [1..30]]
sage: sum([cube((10*a,10*b,10*c), size=random()/3, color=(a,b,c)) for a,b,c in v])
Graphics3d Object

>>> from sage.all import *
>>> v = [(random(), random(), random()) for _ in (ellipsis_range(Integer(1),Ellipsis,Integer(30)))]
>>> sum([cube((Integer(10)*a,Integer(10)*b,Integer(10)*c), size=random()/Integer(3), color=(a,b,c)) for a,b,c in v])
Graphics3d Object
../../../_images/platonic-8.svg

Non-square cubes (boxes):

sage: cube(aspect_ratio=[1,1,1]).scale([1,2,3])
Graphics3d Object

>>> from sage.all import *
>>> cube(aspect_ratio=[Integer(1),Integer(1),Integer(1)]).scale([Integer(1),Integer(2),Integer(3)])
Graphics3d Object
../../../_images/platonic-9.svg
sage: cube(color=['red', 'blue', 'green'],aspect_ratio=[1,1,1]).scale([1,2,3])
Graphics3d Object

>>> from sage.all import *
>>> cube(color=['red', 'blue', 'green'],aspect_ratio=[Integer(1),Integer(1),Integer(1)]).scale([Integer(1),Integer(2),Integer(3)])
Graphics3d Object
../../../_images/platonic-10.svg

And one that is colored:

sage: cube(color=['red', 'blue', 'green', 'black', 'white', 'orange'],
....:      aspect_ratio=[1,1,1]).scale([1,2,3])
Graphics3d Object

>>> from sage.all import *
>>> cube(color=['red', 'blue', 'green', 'black', 'white', 'orange'],
...      aspect_ratio=[Integer(1),Integer(1),Integer(1)]).scale([Integer(1),Integer(2),Integer(3)])
Graphics3d Object
../../../_images/platonic-11.svg

A nice translucent color cube with a frame:

sage: c = cube(color=['red', 'blue', 'green'], frame=False, frame_thickness=2,
....:          frame_color='brown', opacity=0.8)
sage: c
Graphics3d Object

>>> from sage.all import *
>>> c = cube(color=['red', 'blue', 'green'], frame=False, frame_thickness=Integer(2),
...          frame_color='brown', opacity=RealNumber('0.8'))
>>> c
Graphics3d Object
../../../_images/platonic-12.svg

A raytraced color cube with frame and transparency:

sage: c.show(viewer='tachyon')

>>> from sage.all import *
>>> c.show(viewer='tachyon')

This shows Issue #11272 has been fixed:

sage: cube(center=(10, 10, 10), size=0.5).bounding_box()
((9.75, 9.75, 9.75), (10.25, 10.25, 10.25))

>>> from sage.all import *
>>> cube(center=(Integer(10), Integer(10), Integer(10)), size=RealNumber('0.5')).bounding_box()
((9.75, 9.75, 9.75), (10.25, 10.25, 10.25))

AUTHORS:

  • William Stein

sage.plot.plot3d.platonic.dodecahedron(center=(0, 0, 0), size=1, **kwds)[source]#

A dodecahedron.

INPUT:

  • center – (default: (0,0,0))

  • size – (default: 1)

  • color – a string that describes a color; this can also be a list of 3-tuples or strings length 6 or 3, in which case the faces (and oppositive faces) are colored.

  • opacity – (default: 1) if less than 1 then is transparent

EXAMPLES: A plain Dodecahedron:

sage: dodecahedron()
Graphics3d Object

>>> from sage.all import *
>>> dodecahedron()
Graphics3d Object
../../../_images/platonic-13.svg

A translucent dodecahedron that contains a black sphere:

sage: G = dodecahedron(color='orange', opacity=0.8)
sage: G += sphere(size=0.5, color='black')
sage: G
Graphics3d Object

>>> from sage.all import *
>>> G = dodecahedron(color='orange', opacity=RealNumber('0.8'))
>>> G += sphere(size=RealNumber('0.5'), color='black')
>>> G
Graphics3d Object
../../../_images/platonic-14.svg

CONSTRUCTION: This is how we construct a dodecahedron. We let one point be \(Q = (0,1,0)\).

Now there are three points spaced equally on a circle around the north pole. The other requirement is that the angle between them be the angle of a pentagon, namely \(3\pi/5\). This is enough to determine them. Placing one on the \(xz\)-plane we have.

\(P_1 = \left(t, 0, \sqrt{1-t^2}\right)\)

\(P_2 = \left(-\frac{1}{2}t, \frac{\sqrt{3}}{2}t, \sqrt{1-t^2}\right)\)

\(P_3 = \left(-\frac{1}{2}t, \frac{\sqrt{3}}{2}t, \sqrt{1-t^2}\right)\)

Solving \(\frac{(P_1-Q) \cdot (P_2-Q)}{|P_1-Q||P_2-Q|} = \cos(3\pi/5)\) we get \(t = 2/3\).

Now we have 6 points \(R_1, ..., R_6\) to close the three top pentagons. These can be found by mirroring \(P_2\) and \(P_3\) by the \(yz\)-plane and rotating around the \(y\)-axis by the angle \(\theta\) from \(Q\) to \(P_1\). Note that \(\cos(\theta) = t = 2/3\) and so \(\sin(\theta) = \sqrt{5}/3\). Rotation gives us the other four.

Now we reflect through the origin for the bottom half.

AUTHORS:

  • Robert Bradshaw, William Stein

sage.plot.plot3d.platonic.icosahedron(center=(0, 0, 0), size=1, **kwds)[source]#

An icosahedron.

INPUT:

  • center – (default: (0, 0, 0))

  • size – (default: 1)

  • color – a string that describes a color; this can also be a list of 3-tuples or strings length 6 or 3, in which case the faces (and oppositive faces) are colored.

  • opacity – (default: 1) if less than 1 then is transparent

EXAMPLES:

sage: icosahedron()
Graphics3d Object

>>> from sage.all import *
>>> icosahedron()
Graphics3d Object
../../../_images/platonic-15.svg

Two icosahedra at different positions of different sizes.

sage: p = icosahedron((-1/2,0,1), color='orange')
sage: p += icosahedron((2,0,1), size=1/2, color='red', aspect_ratio=[1,1,1])
sage: p
Graphics3d Object

>>> from sage.all import *
>>> p = icosahedron((-Integer(1)/Integer(2),Integer(0),Integer(1)), color='orange')
>>> p += icosahedron((Integer(2),Integer(0),Integer(1)), size=Integer(1)/Integer(2), color='red', aspect_ratio=[Integer(1),Integer(1),Integer(1)])
>>> p
Graphics3d Object
../../../_images/platonic-16.svg
sage.plot.plot3d.platonic.index_face_set(face_list, point_list, enclosed, **kwds)[source]#

Helper function that creates IndexFaceSet object for the tetrahedron, dodecahedron, and icosahedron.

INPUT:

  • face_list – list of faces, given explicitly from the solid invocation

  • point_list – list of points, given explicitly from the solid invocation

  • enclosed – boolean (default passed is always True for these solids)

sage.plot.plot3d.platonic.octahedron(center=(0, 0, 0), size=1, **kwds)[source]#

Return an octahedron.

INPUT:

  • center – (default: (0,0,0))

  • size – (default: 1)

  • color – a string that describes a color; this can also be a list of 3-tuples or strings length 6 or 3, in which case the faces (and oppositive faces) are colored.

  • opacity – (default: 1) if less than 1 then is transparent

EXAMPLES:

sage: G = octahedron((1,4,3), color='orange')
sage: G += octahedron((0,2,1), size=2, opacity=0.6)
sage: G
Graphics3d Object

>>> from sage.all import *
>>> G = octahedron((Integer(1),Integer(4),Integer(3)), color='orange')
>>> G += octahedron((Integer(0),Integer(2),Integer(1)), size=Integer(2), opacity=RealNumber('0.6'))
>>> G
Graphics3d Object
../../../_images/platonic-17.svg
sage.plot.plot3d.platonic.prep(G, center, size, kwds)[source]#

Helper function that scales and translates the platonic solid, and passes extra keywords on.

INPUT:

  • center – 3-tuple indicating the center (default passed from index_face_set() is the origin \((0,0,0)\))

  • size – number indicating amount to scale by (default passed from index_face_set() is 1)

  • kwds – a dictionary of keywords, passed from solid invocation by index_face_set()

sage.plot.plot3d.platonic.tetrahedron(center=(0, 0, 0), size=1, **kwds)[source]#

A 3d tetrahedron.

INPUT:

  • center – (default: (0,0,0))

  • size – (default: 1)

  • color – a string ("red", "green", etc) or a tuple (r, g, b) with r, g, b numbers between 0 and 1

  • opacity – (default: 1) if less than 1 then is transparent

EXAMPLES: A default colored tetrahedron at the origin:

sage: tetrahedron()
Graphics3d Object

>>> from sage.all import *
>>> tetrahedron()
Graphics3d Object
../../../_images/platonic-18.svg

A transparent green tetrahedron in front of a solid red one:

sage: tetrahedron(opacity=0.8, color='green') + tetrahedron((-2,1,0),color='red')
Graphics3d Object

>>> from sage.all import *
>>> tetrahedron(opacity=RealNumber('0.8'), color='green') + tetrahedron((-Integer(2),Integer(1),Integer(0)),color='red')
Graphics3d Object
../../../_images/platonic-19.svg

A translucent tetrahedron sharing space with a sphere:

sage: tetrahedron(color='yellow',opacity=0.7) + sphere(size=.5, color='red')
Graphics3d Object

>>> from sage.all import *
>>> tetrahedron(color='yellow',opacity=RealNumber('0.7')) + sphere(size=RealNumber('.5'), color='red')
Graphics3d Object
../../../_images/platonic-20.svg

A big tetrahedron:

sage: tetrahedron(size=10)
Graphics3d Object

>>> from sage.all import *
>>> tetrahedron(size=Integer(10))
Graphics3d Object
../../../_images/platonic-21.svg

A wide tetrahedron:

sage: tetrahedron(aspect_ratio=[1,1,1]).scale((4,4,1))
Graphics3d Object

>>> from sage.all import *
>>> tetrahedron(aspect_ratio=[Integer(1),Integer(1),Integer(1)]).scale((Integer(4),Integer(4),Integer(1)))
Graphics3d Object
../../../_images/platonic-22.svg

A red and blue tetrahedron touching noses:

sage: tetrahedron(color='red') + tetrahedron((0,0,-2)).scale([1,1,-1])
Graphics3d Object

>>> from sage.all import *
>>> tetrahedron(color='red') + tetrahedron((Integer(0),Integer(0),-Integer(2))).scale([Integer(1),Integer(1),-Integer(1)])
Graphics3d Object
../../../_images/platonic-23.svg

A Dodecahedral complex of 5 tetrahedra (a more elaborate example from Peter Jipsen):

sage: from math import pi
sage: v = (sqrt(5.)/2-5/6, 5/6*sqrt(3.)-sqrt(15.)/2, sqrt(5.)/3)
sage: t = acos(sqrt(5.)/3)/2
sage: t1 = tetrahedron(aspect_ratio=(1,1,1), opacity=0.5).rotateZ(t)
sage: t2 = tetrahedron(color='red', opacity=0.5).rotateZ(t).rotate(v,2*pi/5)
sage: t3 = tetrahedron(color='green', opacity=0.5).rotateZ(t).rotate(v,4*pi/5)
sage: t4 = tetrahedron(color='yellow', opacity=0.5).rotateZ(t).rotate(v,6*pi/5)
sage: t5 = tetrahedron(color='orange', opacity=0.5).rotateZ(t).rotate(v,8*pi/5)
sage: show(t1+t2+t3+t4+t5, frame=False, zoom=1.3)

>>> from sage.all import *
>>> from math import pi
>>> v = (sqrt(RealNumber('5.'))/Integer(2)-Integer(5)/Integer(6), Integer(5)/Integer(6)*sqrt(RealNumber('3.'))-sqrt(RealNumber('15.'))/Integer(2), sqrt(RealNumber('5.'))/Integer(3))
>>> t = acos(sqrt(RealNumber('5.'))/Integer(3))/Integer(2)
>>> t1 = tetrahedron(aspect_ratio=(Integer(1),Integer(1),Integer(1)), opacity=RealNumber('0.5')).rotateZ(t)
>>> t2 = tetrahedron(color='red', opacity=RealNumber('0.5')).rotateZ(t).rotate(v,Integer(2)*pi/Integer(5))
>>> t3 = tetrahedron(color='green', opacity=RealNumber('0.5')).rotateZ(t).rotate(v,Integer(4)*pi/Integer(5))
>>> t4 = tetrahedron(color='yellow', opacity=RealNumber('0.5')).rotateZ(t).rotate(v,Integer(6)*pi/Integer(5))
>>> t5 = tetrahedron(color='orange', opacity=RealNumber('0.5')).rotateZ(t).rotate(v,Integer(8)*pi/Integer(5))
>>> show(t1+t2+t3+t4+t5, frame=False, zoom=RealNumber('1.3'))
../../../_images/platonic-24.svg

AUTHORS:

  • Robert Bradshaw and William Stein