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A [[convex polyhedron]] is said to be composite if there exists a plane through a cycle of its edges that is not a face. Slicing the polyhedron on this plane produces two convex-regular-faced polyhedra, having together the same faces as the original polyhedron, along with two new faces on the plane of the slice.{{r|sod}} Repeated slicing of a polyhedron that cannot produce more convex, regular-faced polyhedra again is called the elementary polyhedron or non-composite polyhedron. One can alternatively define a composite polyhedron as the result of attaching two or more non-composite polyhedra.{{r|timofeenko-2009|hartshorne}} |
A [[convex polyhedron]] is said to be composite if there exists a plane through a cycle of its edges that is not a face. Slicing the polyhedron on this plane produces two convex-regular-faced polyhedra, having together the same faces as the original polyhedron, along with two new faces on the plane of the slice.{{r|sod}} Repeated slicing of a polyhedron that cannot produce more convex, regular-faced polyhedra again is called the elementary polyhedron or non-composite polyhedron. One can alternatively define a composite polyhedron as the result of attaching two or more non-composite polyhedra.{{r|timofeenko-2009|hartshorne}} |
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The [[regular octahedron]] and [[regular icosahedron]] are composite. For a regular octahedron, this can be sliced into two [[equilateral square pyramid]]s, which are elementaries. Slicing the regular icosahedron that removes one and two pyramids by a plane produces other composites, the [[diminished icosahedron]] and [[bidiminished icosahedron]]; removing the third pyramid produces an elementary polyhedron known as [[tridiminished icosahedron |
The [[regular octahedron]] and [[regular icosahedron]] are composite. For a regular octahedron, this can be sliced into two [[equilateral square pyramid]]s, which are elementaries. Slicing the regular icosahedron that removes one and two pyramids by a plane produces other composites, the [[diminished icosahedron]] and [[bidiminished icosahedron]]; removing the third pyramid produces an elementary polyhedron known as [[tridiminished icosahedron]].{{r|timofeenko-2010|hartshorne}} Other examples of elementaries are the [[Prism (geometry)|prism]]s, [[antiprism]]s, and Johnson solids.{{r|timofeenko-2009|johnson}} |
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The regular-faced elementary polyhedra can be enumerated from the convex regular-faced polyhedra. {{harvtxt|Zalgaller|1967}} expressed interest in enumerating the elementary polyhedra whose faces are either regular polyhedra or the sums of regular polygons, providing twenty-eight examples. These are called ”’Zalgaller solids”’.{{r|zalgaller|timofeenko-2010}} {{harvtxt|Ivanov|1971}} and {{harvtxt|Pyrakhin|1973}} provide six more examples, respectively the five ”’Ivanov solids”’ and one ”’Pyrakhin solid”’.{{r|ivanov|pyrakhin|timofeenko-2010}}. |
The regular-faced elementary polyhedra can be enumerated from the convex regular-faced polyhedra. {{harvtxt|Zalgaller|1967}} expressed interest in enumerating the elementary polyhedra whose faces are either regular polyhedra or the sums of regular polygons, providing twenty-eight examples. These are called ”’Zalgaller solids”’.{{r|zalgaller|timofeenko-2010}} {{harvtxt|Ivanov|1971}} and {{harvtxt|Pyrakhin|1973}} provide six more examples, respectively the five ”’Ivanov solids”’ and one ”’Pyrakhin solid”’.{{r|ivanov|pyrakhin|timofeenko-2010}}. |
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Latest revision as of 15:16, 30 November 2025
Polyhedron sliced by a plane into other polyhedra
In geometry, a composite polyhedron is a convex polyhedron that produces two convex, regular-faced polyhedra when sliced by a plane. Repeated slicing of this type until it cannot produce more such polyhedra again is called the elementary polyhedron or non-composite polyhedron.
Definition and examples
[edit]
A convex polyhedron is said to be composite if there exists a plane through a cycle of its edges that is not a face. Slicing the polyhedron on this plane produces two convex-regular-faced polyhedra, having together the same faces as the original polyhedron, along with two new faces on the plane of the slice.[1] Repeated slicing of a polyhedron that cannot produce more convex, regular-faced polyhedra again is called the elementary polyhedron or non-composite polyhedron. One can alternatively define a composite polyhedron as the result of attaching two or more non-composite polyhedra.[2][3]
The regular octahedron and regular icosahedron are composite. For a regular octahedron, this can be sliced into two equilateral square pyramids, which are elementaries. Slicing the regular icosahedron that removes one and two pyramids by a plane produces other composites, the diminished icosahedron and bidiminished icosahedron; removing the third pyramid produces an elementary polyhedron known as tridiminished icosahedron.[4][3] Other examples of elementaries are the prisms, antiprisms, and seventeen Johnson solids (two of which have already been mentioned are the equilateral square pyramid and the tridiminished icosahedron).[2][5]
The regular-faced elementary polyhedra can be enumerated from the convex regular-faced polyhedra. Zalgaller (1967) expressed interest in enumerating the elementary polyhedra whose faces are either regular polyhedra or the sums of regular polygons, providing twenty-eight examples. These are called Zalgaller solids.[6][4] Ivanov (1971) and Pyrakhin (1973) provide six more examples, respectively the five Ivanov solids and one Pyrakhin solid.[7][8][4].
- ^ Slobodan, Mišić; Obradović, Marija; Ðukanović, Gordana (2015). “Composite Concave Cupolae as Geometric and Architectural Forms” (PDF). Journal for Geometry and Graphics. 19 (1): 79–91.
- ^ a b Timofeenko, A. V. (2009). “Convex Polyhedra with Parquet Faces” (PDF). Doklady Mathematics. 80 (2): 720–723. doi:10.1134/S1064562409050238.
- ^ a b Hartshorne, Robin (2000). Geometry: Euclid and Beyond. Undergraduate Texts in Mathematics. Springer-Verlag. p. 464. ISBN 9780387986500.
- ^ a b c Timofeenko, A. V. (2010). “Junction of Non-composite Polyhedra” (PDF). St. Petersburg Mathematical Journal. 21 (3): 483–512. doi:10.1090/S1061-0022-10-01105-2.
- ^ Johnson, Norman (1966). “Convex Solids with Regular Faces”. Canadian Journal of Mathematics. 18: 169–200. doi:10.4153/CJM-1966-021-8.
- ^ Zalgaller, V. A. (1967). “Convex polyhedra with regular faces”. Zapiski Nauchnykh Seminarov LOMI. 2: 5–221.
- ^ Ivanov, B. A. (1971). “Polyhedra with faces that are composed of regular polygons”. Ukrain. Geom. Sb (in Russian) (10): 20–34. MR 0301634.
- ^ Pyrakhin, Yu. A. (1973). “Convex polyhedra with regular faces”. Ukrain. Geom. Sb. (in Russian) (14): 83–88. MR 0338930.
