Coverage for local/lib/python2.7/site-packages/sage/plot/plot3d/platonic.py : 78%

r""" 

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]) 

.. PLOT:: 

G = tetrahedron((0,-3.5,0), color='blue') + cube((0,-2,0),color=(.25,0,.5)) 

G += octahedron(color='red') + dodecahedron((0,2,0), color='orange') 

G += icosahedron(center=(0,4,0), color='yellow') 

sphinx_plot(G) 

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]) 

.. PLOT:: 

G = tetrahedron(color='blue',opacity=0.7) 

G += cube(color=(.25,0,.5), opacity=0.7) 

G += octahedron(color='red', opacity=0.7) 

G += dodecahedron(color='orange', opacity=0.7) + icosahedron(opacity=0.7) 

sphinx_plot(G) 

Display nice faces only:: 

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

Graphics3d Object 

.. PLOT:: 

sphinx_plot(icosahedron().stickers(['red','blue'], .075, .1)) 

AUTHORS: 

- Robert Bradshaw (2007, 2008): initial version 

- William Stein 

""" 

from __future__ import absolute_import 

#***************************************************************************** 

# Copyright (C) 2007 Robert Bradshaw <robertwb@math.washington.edu> 

# 

# Distributed under the terms of the GNU General Public License (GPL) 

# 

# This code is distributed in the hope that it will be useful, 

# but WITHOUT ANY WARRANTY; without even the implied warranty of 

# MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU 

# General Public License for more details. 

# 

# The full text of the GPL is available at: 

# 

# http://www.gnu.org/licenses/ 

#***************************************************************************** 

from sage.rings.all import RDF 

from sage.matrix.constructor import matrix 

from .shapes import Box, ColorCube 

from .shapes2 import frame3d 

from .index_face_set import IndexFaceSet 

def index_face_set(face_list, point_list, enclosed, **kwds): 

""" 

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) 

TESTS: 

Verify that these are working and passing on keywords:: 

sage: tetrahedron(center=(2,0,0),size=2,color='red') 

Graphics3d Object 

:: 

sage: dodecahedron(center=(2,0,0),size=2,color='red') 

Graphics3d Object 

:: 

sage: icosahedron(center=(2,0,0),size=2,color='red') 

Graphics3d Object 

""" 

if 'center' in kwds: 

center = kwds['center'] 

del kwds['center'] 

else: 

center = (0, 0, 0) 

if 'size' in kwds: 

size = kwds['size'] 

del kwds['size'] 

else: 

size = 1 

I = IndexFaceSet(face_list, point_list, enclosed=enclosed, **kwds) 

return prep(I, center, size, kwds) 

def prep(G, center, size, kwds): 

""" 

Helper function that scales and translates the platonic 

solid, and passes extra keywords on. 

INPUT: 

- ``center`` -- 3-tuple indicating the center (default passed 

from :func:`index_face_set` is the origin `(0,0,0)`) 

- ``size`` -- number indicating amount to scale by (default 

passed from :func:`index_face_set` is 1) 

- ``kwds`` -- a dictionary of keywords, passed from solid 

invocation by :func:`index_face_set` 

TESTS: 

Verify that scaling and moving the center work together properly, 

and that keywords are passed (see :trac:`10796`):: 

sage: octahedron(center=(2,0,0),size=2,color='red') 

Graphics3d Object 

""" 

if size != 1: 

G = G.scale(size) 

if center != (0, 0, 0): 

G = G.translate(center) 

G._set_extra_kwds(kwds) 

return G 

def tetrahedron(center=(0, 0, 0), size=1, **kwds): 

""" 

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 

.. PLOT:: 

sphinx_plot(tetrahedron()) 

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 

.. PLOT:: 

sphinx_plot(tetrahedron(opacity=0.8, color='green') + tetrahedron((-2,1,0),color='red')) 

A translucent tetrahedron sharing space with a sphere:: 

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

Graphics3d Object 

.. PLOT:: 

sphinx_plot(tetrahedron(color='yellow',opacity=0.7) + sphere(size = .5, color='red')) 

A big tetrahedron:: 

sage: tetrahedron(size=10) 

Graphics3d Object 

.. PLOT:: 

sphinx_plot(tetrahedron(size=10)) 

A wide tetrahedron:: 

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

Graphics3d Object 

.. PLOT:: 

sphinx_plot(tetrahedron(aspect_ratio=[1,1,1]).scale((4,4,1))) 

A red and blue tetrahedron touching noses:: 

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

Graphics3d Object 

.. PLOT:: 

sphinx_plot(tetrahedron(color='red') + tetrahedron((0,0,-2)).scale([1,1,-1])) 

A Dodecahedral complex of 5 tetrahedra (a more elaborate example 

from Peter Jipsen):: 

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) 

.. PLOT:: 

v=(sqrt(5.)/2-5/6, 5/6*sqrt(3.)-sqrt(15.)/2, sqrt(5.)/3) 

t=acos(sqrt(5.)/3)/2 

t1=tetrahedron(aspect_ratio=(1,1,1), opacity=0.5).rotateZ(t) 

t2=tetrahedron(color='red', opacity=0.5).rotateZ(t).rotate(v,2*pi/5) 

t3=tetrahedron(color='green', opacity=0.5).rotateZ(t).rotate(v,4*pi/5) 

t4=tetrahedron(color='yellow', opacity=0.5).rotateZ(t).rotate(v,6*pi/5) 

t5=tetrahedron(color='orange', opacity=0.5).rotateZ(t).rotate(v,8*pi/5) 

sphinx_plot(t1+t2+t3+t4+t5) 

AUTHORS: 

- Robert Bradshaw and William Stein 

""" 

RR = RDF 

one = RR.one() 

sqrt2 = RR(2).sqrt() 

sqrt6 = RR(6).sqrt() 

point_list = [(0,0,1), 

(2*sqrt2/3, 0, -one/3), 

( -sqrt2/3, sqrt6/3, -one/3), 

( -sqrt2/3, -sqrt6/3, -one/3)] 

face_list = [[0,1,2],[1,3,2],[0,2,3],[0,3,1]] 

if 'aspect_ratio' not in kwds: 

kwds['aspect_ratio'] = [1, 1, 1] 

return index_face_set(face_list, point_list, enclosed=True, center=center, size=size, **kwds) 

def cube(center=(0, 0, 0), size=1, color=None, frame_thickness=0, 

frame_color=None, **kwds): 

""" 

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 

.. PLOT:: 

sphinx_plot(cube()) 

A red cube:: 

sage: cube(color="red") 

Graphics3d Object 

.. PLOT:: 

sphinx_plot(cube(color="red")) 

A transparent grey cube that contains a red cube:: 

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

Graphics3d Object 

.. PLOT:: 

sphinx_plot(cube(opacity=0.8, color='grey') + cube(size=3/4)) 

A transparent colored cube:: 

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

Graphics3d Object 

.. PLOT:: 

sphinx_plot(cube(color=['red', 'green', 'blue'], opacity=0.5)) 

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 

.. PLOT:: 

v = [(random(), random(), random()) for _ in range(30)] 

sphinx_plot(sum([cube((10*a,10*b,10*c), size=random()/3, color=(a,b,c)) for a,b,c in v])) 

Non-square cubes (boxes):: 

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

Graphics3d Object 

.. PLOT:: 

sphinx_plot(cube(aspect_ratio=[1,1,1]).scale([1,2,3])) 

:: 

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

Graphics3d Object 

.. PLOT:: 

sphinx_plot(cube(color=['red', 'blue', 'green'],aspect_ratio=[1,1,1]).scale([1,2,3])) 

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 

.. PLOT:: 

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

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 

.. PLOT:: 

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

sphinx_plot(c) 

A raytraced color cube with frame and transparency:: 

sage: c.show(viewer='tachyon') 

This shows :trac:`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)) 

AUTHORS: 

- William Stein 

""" 

if isinstance(color, (list, tuple)) and len(color) > 0 and isinstance(color[0], (list,tuple,str)): 

B = ColorCube(size=[0.5,0.5,0.5], colors=color, **kwds) 

else: 

if color is not None: 

kwds['color'] = color 

B = Box(0.5, 0.5, 0.5, **kwds) 

if frame_thickness > 0: 

if frame_color is None: 

B += frame3d((-0.5,-0.5,-0.5),(0.5,0.5,0.5), thickness=frame_thickness) 

else: 

B += frame3d((-0.5,-0.5,-0.5),(0.5,0.5,0.5), thickness=frame_thickness, color=frame_color) 

return prep(B, center, size, kwds) 

def octahedron(center=(0, 0, 0), size=1, **kwds): 

r""" 

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 

.. PLOT:: 

G = octahedron((1,4,3), color='orange') 

G += octahedron((0,2,1), size=2, opacity=0.6) 

sphinx_plot(G) 

""" 

kwds['enclosed'] = True 

if 'aspect_ratio' not in kwds: 

kwds['aspect_ratio'] = [1, 1, 1] 

return prep(Box(1,1,1).dual(**kwds), center, size, kwds) 

def dodecahedron(center=(0, 0, 0), size=1, **kwds): 

r""" 

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 

.. PLOT:: 

sphinx_plot(dodecahedron()) 

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 

.. PLOT:: 

G = dodecahedron(color='orange', opacity=0.8) 

G += sphere(size=0.5, color='black') 

sphinx_plot(G) 

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 

""" 

RR = RDF 

one = RR(1) 

sqrt3 = RR(3).sqrt(); 

sqrt5 = RR(5).sqrt() 

R3 = RR**3 

rot = matrix(RR, [[ -one/2,-sqrt3/2, 0], 

[ sqrt3/2, -one/2, 0], 

[ 0, 0, 1]]) 

rot2 = rot*rot 

# The top 

Q = R3([0,0,1]) 

# The first ring 

P1 = R3([2*one/3, 0, sqrt5/3]) 

# The second ring 

R1 = R3([sqrt5/3, 1/sqrt3, one/3]) 

R2 = R3([sqrt5/3, -1/sqrt3, one/3]) 

top = [Q, P1, rot*P1, rot2*P1, R1, rot*R2, rot*R1, rot2*R2, rot2*R1, R2] 

point_list = top + [-p for p in reversed(top)] 

top_faces = [[0,1,4,5,2], 

[0,2,6,7,3], 

[0,3,8,9,1], 

[1,9,13,12,4], 

[2,5,11,10,6], 

[3,7,15,14,8]] 

face_list = top_faces + [[19-p for p in reversed(f)] for f in top_faces] 

if 'aspect_ratio' not in kwds: 

kwds['aspect_ratio'] = [1,1,1] 

return index_face_set(face_list, point_list, enclosed=True, center=center, size=size, **kwds) 

# if style == 'vertices' or style == 'edges': 

# from sage.plot.colors import rainbow 

# colors = rainbow(len(vs), 'rgbtuple') 

# #vertex_spheres = [Box(.05, .05, .05, color=color).translate(p) for p in vs] 

# vertex_spheres = [Box(.05, .05, .05, color=c).translate(p) for p,c in zip(vs,colors)] 

# faces = IndexFaceSet([[tuple(vs[i]) for i in f] for f in face_list]) 

# vertex_spheres += [faces.stickers(['red','yellow','blue','purple','black','orange'], .1, .1)] # [faces] 

# return Graphics3dGroup(vertex_spheres) 

def icosahedron(center=(0, 0, 0), size=1, **kwds): 

r""" 

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 

.. PLOT:: 

sphinx_plot(icosahedron()) 

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 

.. PLOT:: 

p = icosahedron((-1/2,0,1), color='orange') 

p += icosahedron((2,0,1), size = 0.5, color='red', aspect_ratio=[1,1,1]) 

sphinx_plot(p) 

""" 

kwds['enclosed'] = True 

if 'aspect_ratio' not in kwds: 

kwds['aspect_ratio'] = [1, 1, 1] 

return prep(dodecahedron().dual(**kwds), center, size, kwds)