Regular icosahedron
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Regular icosahedron | |
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Type | Gyroelongated bipyramid Deltahedron |
Faces | 20 |
Edges | 30 |
Vertices | 12 |
Vertex configuration | |
Symmetry group | Icosahedral symmetry |
Dihedral angle (degrees) | 138.190 (approximately) |
Dual polyhedron | Regular dodecahedron |
Net | |
In
Many polyhedrons are constructed from the regular icosahedron. For example, most of the Kepler–Poinsot polyhedron is constructed by faceting. Some of the Johnson solids can be constructed by removing the pentagonal pyramids. The regular icosahedron has many relations with other Platonic solids, one of them is the regular dodecahedron as its dual polyhedron and has the historical background on the comparison mensuration. It also has many relations with other polytopes.
The appearance of regular icosahedron can be found in nature, such as the virus with icosahedral-shaped
Construction
The regular icosahedron can be constructed like other
Another way to construct it is by putting two points on each surface of a cube. In each face, draw a segment line between the midpoints of two opposite edges and locate two points with the golden ratio distance from each midpoint. These twelve vertices describe the three mutually perpendicular planes, with edges drawn between each of them.[4] Because of the constructions above, the regular icosahedron is Platonic solid, a family of polyhedra with regular faces. A polyhedron with only equilateral triangles as faces is called a deltahedron. There are only eight different convex deltahedra, one of which is the regular icosahedron.[5]
One possible system of
Properties
Mensuration
The
The surface area of polyhedra is the sum of its every face. Therefore, the surface area of regular icosahedra equals the area of 20 equilateral triangles. The volume of a regular icosahedron is obtained by calculating the volume of all pyramids with the base of triangular faces and the height with the distance from a triangular face's centroid to the center inside the regular icosahedron, the circumradius of a regular icosahedron.[8]
The dihedral angle of a regular icosahedron can be calculated by adding the angle of pentagonal pyramids with regular faces and a pentagonal antiprism. The dihedral angle of a pentagonal antiprism and pentagonal pyramid between two adjacent triangular faces are approximately . The dihedral angle of a pentagonal antiprism between pentagon-to-triangle is , and the dihedral angle of a pentagonal pyramid between the same faces is . Therefore, for the regular icosahedron, the dihedral angle between two adjacent triangles, on the edge where the pentagonal pyramid and pentagonal antiprism are attached is .[13]
Symmetry
The rotational
The full symmetry group of the icosahedron (including reflections) is known as the
Icosahedral graph
Every
The icosahedral graph is
Related polyhedra
In other Platonic solids
Aside from comparing the mensuration between the regular icosahedron and regular dodecahedron, they are dual to each other. An icosahedron can be inscribed in a dodecahedron by placing its vertices at the face centers of the dodecahedron, and vice versa.[17]
An icosahedron can be inscribed in an
An icosahedron of edge length can be inscribed in a unit-edge-length cube by placing six of its edges (3 orthogonal opposite pairs) on the square faces of the cube, centered on the face centers and parallel or perpendicular to the square's edges.[19] Because there are five times as many icosahedron edges as cube faces, there are five ways to do this consistently, so five disjoint icosahedra can be inscribed in each cube. The edge lengths of the cube and the inscribed icosahedron are in the golden ratio.[20]
Stellation
The icosahedron has a large number of
The faces of the icosahedron extended outwards as planes intersect, defining regions in space as shown by this stellation diagram of the intersections in a single plane. |
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Facetings
The
Diminishment
The
Relations to the 600-cell and other 4-polytopes
The icosahedron is the dimensional analogue of the 600-cell, a regular 4-dimensional polytope. The 600-cell has icosahedral cross sections of two sizes, and each of its 120 vertices is an icosahedral pyramid; the icosahedron is the vertex figure of the 600-cell.
The unit-radius 600-cell has tetrahedral cells of edge length , 20 of which meet at each vertex to form an icosahedral pyramid (a
A semiregular 4-polytope, the snub 24-cell, has icosahedral cells.
Relations to other uniform polytopes
As has been mentioned above, the regular icosahedron is unique among the
There are distortions of the icosahedron that, while no longer regular, are nevertheless
Relation to the 6-cube and rhombic triacontahedron
The icosahedron can be projected to 3D from the 6D 6-demicube using the same basis vectors that form the hull of the rhombic triacontahedron from the 6-cube. Shown here including the inner 20 vertices which are not connected by the 30 outer hull edges of 6D norm length . The inner vertices form a dodecahedron.
The 3D projection basis vectors [u,v,w] used are:
Applications and its natural form
Dice are the common objects with the different polyhedron, one of them is the regular icosahedron. The twenty-sided dice was found in many ancient times. One example is the dice from the Ptolemaic of Egypt, which was later the Greek letters inscribed on the faces in the period of Greece and Roman.[22] Another example was found in the treasure of
The regular icosahedron may also appear in many fields of science. In
Notes
- ^ See Jones 2003 for the pronunciation: (/ˌaɪkɒsəˈhiːdrən, -kə-, -koʊ-/ or /aɪˌkɒsəˈhiːdrən/)
- ^ Silvester 2001, p. 140–141; Cundy 1952.
- ^ a b Berman 1971.
- ^ Cromwell 1997, p. 70.
- ^ Shavinina 2013, p. 333; Cundy 1952.
- ^ Steeb, Hardy & Tanski 2012, p. 211.
- ^ MacLean 2007, p. 43–44; Coxeter 1973, Table I(i), pp. 292–293. See column "", where is Coxeter's notation for the midradius, also noting that Coxeter uses as the edge length (see p. 2).
- ^ MacLean 2007, p. 43–44.
- ^ Herz-Fischler 2013, p. 138–140.
- ^ Simmons 2007, p. 50.
- ^ Sutton 2002, p. 55.
- ^ Numerical values for the volumes of the inscribed Platonic solids may be found in Buker & Eggleton 1969.
- ^ Johnson 1966, See table II, line 4.; MacLean 2007, p. 43–44.
- ^ Klein 1884. See icosahedral symmetry: related geometries for further history, and related symmetries on seven and eleven letters.
- ^ Bickle 2020, p. 147.
- ^ Hopkins 2004.
- ^ Herrmann & Sally 2013, p. 257.
- ^ Coxeter et al. 1938, p. 4.
- ^ Borovik 2006, pp. 8–9, §5. How to draw an icosahedron on a blackboard.
- ^ Reciprocally, the edge length of a cube inscribed in a dodecahedron is in the golden ratio to the dodecahedron's edge length. The cube's edges lie in pentagonal face planes of the dodecahedron as regular pentagon diagonals, which are always in the golden ratio to the regular pentagon's edge. When a cube is inscribed in a dodecahedron and an icosahedron is inscribed in the cube, the dodecahedron and icosahedron that do not share any vertices have the same edge length.
- ^ Coxeter et al. 1938, p. 8–26.
- ^ Smith 1958, p. 295; Minas-Nerpel 2007.
- ^ Cromwell 1997, p. 4.
- ^ "Dungeons & Dragons Dice". gmdice.com. Retrieved August 20, 2019.
- ^ Flanagan & Gregory 2015, p. 85.
- ^ Strauss & Strauss 2008, p. 35–62.
- ^ Haeckel 1904; Cromwell 1997, p. 6.
- ^ Spokoyny 2013.
- ^ Hofmeister 2004.
- ^ Dronskowski, Kikkawa & Stein 2017, p. 435–436.
- ^ Cromwell 1997, p. 7.
- ^ Whyte 1952.
References
- Berman, Martin (1971). "Regular-faced convex polyhedra". Journal of the Franklin Institute. 291 (5): 329–352. MR 0290245.
- Bickle, Allan (2020). Fundamentals of Graph Theory. ISBN 9781470455491.
- ISBN 978-0821837221.
- Buker, W. E.; Eggleton, R. B. (1969). "The Platonic Solids (Solution to problem E2053)". JSTOR 2317282.
- MR 0370327.
- Coxeter, H.S.M.; du Val, Patrick; Flather, H. T.; Petrie, J. F. (1938). The Fifty-Nine Icosahedra. Vol. 6. University of Toronto Studies (Mathematical Series).
- Cromwell, Peter R. (1997). Polyhedra. ISBN 978-0-521-55432-9.
- Cundy, H. Martyn (1952). "Deltahedra". The Mathematical Gazette. 36 (318): 263–266. S2CID 250435684.
- Dronskowski, Richard; Kikkawa, Shinichi; Stein, Andreas (2017). Handbook of Solid State Chemistry, 6 Volume Set. John Sons & Wiley. ISBN 978-3-527-69103-6.
- Flanagan, Kieran; Gregory, Dan (2015). Selfish, Scared and Stupid: Stop Fighting Human Nature and Increase Your Performance, Engagement and Influence. ISBN 9780730312796.
- Haeckel, E. (1904). Kunstformen der Natur (in German). See here for an online book.
- Herrmann, Diane L.; Sally, Paul J. (2013). Number, Shape, & Symmetry: An Introduction to Number Theory, Geometry, and Group Theory. CRC Press. ISBN 978-1-4665-5464-1.
- Herz-Fischler, Roger (2013). A Mathematical History of the Golden Number. Courier Dover Publications. ISBN 9780486152325.
- Hofmeister, H. (2004). "Fivefold Twinned Nanoparticles". Encyclopedia of Nanoscience and Nanotechnology. 3: 431–452.
- Hopkins, Brian (2004). "Hamiltonian paths on Platonic graphs". .
- Zbl 0132.14603.
- ISBN 3-12-539683-2.
- ISBN 978-0-486-49528-6, Dover edition. Teubner.
{{cite book}}
: CS1 maint: postscript (link), translated from Klein, Felix (1884). Vorlesungen über das Ikosaeder und die Auflösung der Gleichungen vom fünften Grade - MacLean, Kenneth J. M. (2007). A Geometric Analysis of the Platonic Solids and Other Semi-Regular Polyhedra. Loving Healing Press. ISBN 978-1-932690-99-6.
- Minas-Nerpel, Martina (2007). "A Demotic Inscribed Icosahedron from Dakhleh Oasis". The Journal of Egyptian Archaeology. 93: 137–148. JSTOR 40345834.
- Shavinina, Larisa V. (2013). The Routledge International Handbook of Innovation Education. Routledge. ISBN 978-0-203-38714-6.
- Silvester, John R. (2001). Geometry: Ancient and Modern. Oxford University Publisher.
- Simmons, George F. (2007). Calculus Gems: Brief Lives and Memorable Mathematics. Mathematical Association of America. ISBN 9780883855614.
- Smith, David E. (1958). History of Mathematics. Vol. 2. Dover Publications. ISBN 0-486-20430-8.
- PMID 24311823.
- Steinmitz, Nicole F.; Manchester, Marianne (2011). Viral Nanoparticles: Tools for Material Science and Biomedicine. Pan Stanford Publisher. ISBN 978-981-4267-94-6.
- Strauss, James H.; Strauss, Ellen G. (2008). "The Structure of Viruses". Viruses and Human Disease. Elsevier. S2CID 80803624.
- Sutton, Daud (2002). Platonic & Archimedean Solids. Wooden Books. Bloomsbury Publishing USA. ISBN 9780802713865.
- Steeb, Willi-hans; Hardy, Yorick; Tanski, Igor (2012). Problems And Solutions For Groups, Lie Groups, Lie Algebras With Applications. World Scientific Publishing Company. ISBN 9789813104112.
- Whyte, L. L. (1952). "Unique arrangements of points on a sphere". MR 0050303.
External links
- Klitzing, Richard. "3D convex uniform polyhedra x3o5o – ike".
- Hartley, Michael. "Dr Mike's Math Games for Kids".
- K.J.M. MacLean, A Geometric Analysis of the Five Platonic Solids and Other Semi-Regular Polyhedra
- Virtual Reality Polyhedra The Encyclopedia of Polyhedra
- Tulane.edu A discussion of viral structure and the icosahedron
- Origami Polyhedra – Models made with Modular Origami
- Video of icosahedral mirror sculpture
- [1] Principle of virus architecture
Family | An | Bn | I2(p) / Dn | E6 / E7 / E8 / F4 / G2 | Hn
| |||||||
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Regular polygon | Triangle | Square | p-gon | Hexagon | Pentagon | |||||||
Uniform polyhedron | Tetrahedron | Octahedron • Cube | Demicube | Dodecahedron • Icosahedron | ||||||||
Uniform polychoron
|
Pentachoron | 16-cell • Tesseract | Demitesseract | 24-cell | 120-cell • 600-cell | |||||||
Uniform 5-polytope | 5-simplex | 5-orthoplex • 5-cube | 5-demicube | |||||||||
Uniform 6-polytope | 6-simplex | 6-orthoplex • 6-cube | 6-demicube | 122 • 221 | ||||||||
Uniform 7-polytope | 7-simplex | 7-orthoplex • 7-cube | 7-demicube | 132 • 231 • 321 | ||||||||
Uniform 8-polytope | 8-simplex | 8-orthoplex • 8-cube | 8-demicube | 142 • 241 • 421 | ||||||||
Uniform 9-polytope | 9-simplex | 9-orthoplex • 9-cube | 9-demicube | |||||||||
Uniform 10-polytope | 10-simplex | 10-orthoplex • 10-cube | 10-demicube | |||||||||
Uniform n-polytope | n-simplex | n-orthoplex • n-cube | n-demicube | 1k2 • 2k1 • k21 | n-pentagonal polytope | |||||||
Topics: List of regular polytopes and compounds
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Notable stellations of the icosahedron | |||||||||
Regular | Uniform duals | Regular compounds
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Regular star | Others | |||||
(Convex) icosahedron | Small triambic icosahedron | Medial triambic icosahedron
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Great triambic icosahedron | Compound of five octahedra | Compound of five tetrahedra | Compound of ten tetrahedra | Great icosahedron | Excavated dodecahedron | Final stellation |
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The stellation process on the icosahedron creates a number of related compounds with icosahedral symmetry .
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