WOLFRAM

Wolfram Language & System Documentation Center
Wolfram Language Home Page »

Polyhedron
Copy to clipboard.
✖
Polyhedron

Copy to clipboard.
✖
Polyhedron[{f1,…,fn}]

represents a filled polyhedron inside the closed surfaces with polygon faces fi.

Copy to clipboard.
✖
Polyhedron[{f1,…,fn}{{g1,…,gm},…}]

represents a polyhedron with voids {g1,…,gm},….

Copy to clipboard.
✖
Polyhedron[{poly1,poly2,…}]

represents a collection of polyhedra polyi.

Copy to clipboard.
✖
Polyhedron[{p1,…,pn},data]

represents a polyhedron in which coordinates given as integers i in data are taken to be pi.

Details and Options

  • Polyhedron can be used a geometric region and a graphics primitive.
  • Polyhedron[{f1,…,fn}] is a volume region, representing all the points inside the closed surface with polygon faces fi.
  • A point is an element of the polyhedron if a ray from the point in any direction crosses the boundary polygon faces an odd number of times.
  • Polyhedron[{f1,…,fn}{{g1,…,gm},…}] specifies a polyhedron with voids consisting of an outer polyhedron Polyhedron[{f1,…,fn}] and one or several inner polyhedra Polyhedron[{g1,…,gm}],….
  • A point p is an element of the polyhedron if it is in the outer polyhedron but not in any inner polyhedron.
  • Polyhedron[{poly1,poly2,…}] is a collection of polyhedra polyi with or without voids and is treated as a union of polyi for geometric computations.
  • Polyhedron[{p1,…,pn},data] effectively replaces integers i that appear as coordinates in data by the corresponding pi.
  • Polyhedron[{p1,…,pn},{f1,…,fn}]polyhedron boundary faces fi with points {po1,…,pok}
    Polyhedron[{p1,…,pn},{{f1,…,fk}{{g1,…,gl},…}]outer polyhedron boundary faces fi with points {po1,…,pok} and inner polyhedron boundary faces gj with points {pv1,…,pvl} etc.
    Polyhedron[{p1,…,pn},{{b1,…,bn},{f1,…,fk}{{g1,…,gl},…},…}]a collection of several polyhedra
  • As a geometric region, the polygon faces fi can have any embedding dimension, but must all be simple polygons and have the same embedding dimension.
  • In a graphics, the points of the polygon faces fi can be Scaled and Dynamic expressions.
  • Graphics renderings is affected by directives such as FaceForm, EdgeForm, Texture, Specularity, Opacity and color.
  • The following options and settings can be used in graphics:
  • VertexColorsAutomaticvertex colors to be interpolated
    VertexNormalsAutomaticeffective vertex normals for shading
    VertexTextureCoordinatesNonecoordinates for textures

Examples

open allclose all

Basic Examples  (1)Summary of the most common use cases

A polyhedron:

Copy to clipboard.
In[8]:=8
✖
https://wolfram.com/xid/0fq5vjr8bn06a-y1y428
Direct link to example
Out[8]=8

Its graphic image:

Copy to clipboard.
In[9]:=9
✖
https://wolfram.com/xid/0fq5vjr8bn06a-d9xwqy
Direct link to example
Out[9]=9

Its volume:

Copy to clipboard.
In[10]:=10
✖
https://wolfram.com/xid/0fq5vjr8bn06a-puccu
Direct link to example
Out[10]=10

Scope  (11)Survey of the scope of standard use cases

Graphics  (8)

Specification  (2)

Polyhedra:

Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0fq5vjr8bn06a-40503z
Direct link to example
Out[1]=1

Polyhedra with voids:

Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0fq5vjr8bn06a-dzz70h
Direct link to example
Out[1]=1

Styling  (5)

Color directives specify the face colors of polyhedra:

Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0fq5vjr8bn06a-dhl178
Direct link to example
Out[3]=3

Texture can be used to specify a texture to be used on the faces of polyhedra:

Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0fq5vjr8bn06a-i3gj8t
Direct link to example
Out[1]=1

Texture can work together with a different Opacity:

Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0fq5vjr8bn06a-dzpxme
Direct link to example
Out[2]=2

Texture can work together with different Lighting:

Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0fq5vjr8bn06a-mfzdza
Direct link to example
Out[3]=3

FaceForm and EdgeForm can be used to specify the styles of the edges and faces:

Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0fq5vjr8bn06a-hg4riy
Direct link to example
Out[1]=1

Colors can be specified at vertices using VertexColors:

Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0fq5vjr8bn06a-cn0av
Direct link to example
Out[1]=1

Normals can be specified at vertices using VertexNormals for polyhedra:

Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0fq5vjr8bn06a-pil8cd
Direct link to example
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0fq5vjr8bn06a-jkefut
Direct link to example
Out[2]=2

Coordinates  (1)

Use Scaled coordinates:

Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0fq5vjr8bn06a-7wd43
Direct link to example
Out[1]=1

Regions  (3)

Embedding dimension:

Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0fq5vjr8bn06a-y220
Direct link to example
Out[1]=1

Geometric dimension:

Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0fq5vjr8bn06a-bx9tom
Direct link to example
Out[2]=2

Volume:

Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0fq5vjr8bn06a-kdm7fz
Direct link to example
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0fq5vjr8bn06a-epfmzd
Direct link to example
Out[2]=2

Centroid:

Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0fq5vjr8bn06a-dxlbau
Direct link to example
Out[3]=3
Copy to clipboard.
In[4]:=4
✖
https://wolfram.com/xid/0fq5vjr8bn06a-dvbndl
Direct link to example
Out[4]=4

A polyhedron is bounded:

Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0fq5vjr8bn06a-pjgwn2
Direct link to example
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0fq5vjr8bn06a-b5d6xy
Direct link to example
Out[2]=2

Find its range:

Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0fq5vjr8bn06a-jvsvvm
Direct link to example
Out[3]=3
Copy to clipboard.
In[4]:=4
✖
https://wolfram.com/xid/0fq5vjr8bn06a-g9tncx
Direct link to example
Out[4]=4

Possible Issues  (1)Common pitfalls and unexpected behavior

Degenerate polyhedra are not valid geometric regions:

Copy to clipboard.
In[5]:=5
✖
https://wolfram.com/xid/0fq5vjr8bn06a-tw52q1
Direct link to example
Out[5]=5
Copy to clipboard.
In[6]:=6
✖
https://wolfram.com/xid/0fq5vjr8bn06a-e710uk
Direct link to example
Out[6]=6
Wolfram Research (2019), Polyhedron, Wolfram Language function, https://reference.wolfram.com/language/ref/Polyhedron.html.
Copy to clipboard.
✖
Wolfram Research (2019), Polyhedron, Wolfram Language function, https://reference.wolfram.com/language/ref/Polyhedron.html.

Text

Wolfram Research (2019), Polyhedron, Wolfram Language function, https://reference.wolfram.com/language/ref/Polyhedron.html.

Copy to clipboard.
✖
Wolfram Research (2019), Polyhedron, Wolfram Language function, https://reference.wolfram.com/language/ref/Polyhedron.html.

CMS

Wolfram Language. 2019. "Polyhedron." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/Polyhedron.html.

Copy to clipboard.
✖
Wolfram Language. 2019. "Polyhedron." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/Polyhedron.html.

APA

Wolfram Language. (2019). Polyhedron. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/Polyhedron.html

Copy to clipboard.
✖
Wolfram Language. (2019). Polyhedron. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/Polyhedron.html

BibTeX

@misc{reference.wolfram_2025_polyhedron, author="Wolfram Research", title="{Polyhedron}", year="2019", howpublished="\url{https://reference.wolfram.com/language/ref/Polyhedron.html}", note=[Accessed: 04-April-2025 ]}

Copy to clipboard.
✖
@misc{reference.wolfram_2025_polyhedron, author="Wolfram Research", title="{Polyhedron}", year="2019", howpublished="\url{https://reference.wolfram.com/language/ref/Polyhedron.html}", note=[Accessed: 04-April-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_polyhedron, organization={Wolfram Research}, title={Polyhedron}, year={2019}, url={https://reference.wolfram.com/language/ref/Polyhedron.html}, note=[Accessed: 04-April-2025 ]}

Copy to clipboard.
✖
@online{reference.wolfram_2025_polyhedron, organization={Wolfram Research}, title={Polyhedron}, year={2019}, url={https://reference.wolfram.com/language/ref/Polyhedron.html}, note=[Accessed: 04-April-2025 ]}