In
geometryGeometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....
, a
Platonic solid is a
convexA convex polytope is a special case of a polytope, having the additional property that it is also a convex set of points in the n-dimensional space Rn...
polyhedronIn elementary geometry a polyhedron is a geometric solid in three dimensions with flat faces and straight edges...
that is
regularA regular polyhedron is a polyhedron whose faces are congruent regular polygons which are assembled in the same way around each vertex. A regular polyhedron is highly symmetrical, being all of edge-transitive, vertex-transitive and face-transitive - i.e. it is transitive on its flags...
, in the sense of a
regular polygonA regular polygon is a polygon that is equiangular and equilateral . Regular polygons may be convex or star.-General properties:...
. Specifically, the faces of a Platonic solid are
congruentIn geometry, two figures are congruent if they have the same shape and size. This means that either object can be repositioned so as to coincide precisely with the other object...
regular polygons, with the same number of faces meeting at each
vertexIn geometry, a vertex is a special kind of point that describes the corners or intersections of geometric shapes.-Of an angle:...
; thus, all its edges are congruent, as are its vertices and angles.
There are five Platonic solids (shown below):
TetrahedronIn geometry, a tetrahedron is a polyhedron composed of four triangular faces, three of which meet at each vertex. A regular tetrahedron is one in which the four triangles are regular, or "equilateral", and is one of the Platonic solids...
|
CubeIn geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. The cube can also be called a regular hexahedron and is one of the five Platonic solids. It is a special kind of square prism, of rectangular parallelepiped and...
(or Regular hexahedronA hexahedron is any polyhedron with six faces, although usually implies the cube as a regular hexahedron with all its faces square, and three squares around each vertex....
) |
OctahedronIn geometry, an octahedron is a polyhedron with eight faces. A regular octahedron is a Platonic solid composed of eight equilateral triangles, four of which meet at each vertex....
|
Dodecahedron |
IcosahedronIn geometry, an icosahedron is a regular polyhedron with 20 identical equilateral triangular faces, 30 edges and 12 vertices. It is one of the five Platonic solids....
|
(Animation) |
(Animation) |
(Animation) |
(Animation) |
(Animation) |
The name of each figure is derived from its number of faces: respectively 4, 6, 8, 12, and 20.
The
aesthetic beautyMany mathematicians derive aesthetic pleasure from their work, and from mathematics in general. They express this pleasure by describing mathematics as beautiful. Sometimes mathematicians describe mathematics as an art form or, at a minimum, as a creative activity...
and
symmetrySymmetry generally conveys two primary meanings. The first is an imprecise sense of harmonious or aesthetically pleasing proportionality and balance; such that it reflects beauty or perfection...
of the Platonic solids have made them a favorite subject of geometers for thousands of years. They are named for the
ancient Greek philosopherAncient Greek philosophy arose in the 6th century BCE and continued through the Hellenistic period, at which point Ancient Greece was incorporated in the Roman Empire...
PlatoPlato , was a Classical Greek philosopher, mathematician, student of Socrates, writer of philosophical dialogues, and founder of the Academy in Athens, the first institution of higher learning in the Western world. Along with his mentor, Socrates, and his student, Aristotle, Plato helped to lay the...
who theorized that the
classical elementMany philosophies and worldviews have a set of classical elements believed to reflect the simplest essential parts and principles of which anything consists or upon which the constitution and fundamental powers of anything are based. Most frequently, classical elements refer to ancient beliefs...
s were constructed from the regular solids. With
dualityIn geometry, polyhedra are associated into pairs called duals, where the vertices of one correspond to the faces of the other. The dual of the dual is the original polyhedron. The dual of a polyhedron with equivalent vertices is one with equivalent faces, and of one with equivalent edges is another...
,
truncationIn geometry, a truncation is an operation in any dimension that cuts polytope vertices, creating a new facet in place of each vertex.- Uniform truncation :...
, and
snubbingIn geometry, an alternation is an operation on a polyhedron or tiling that removes alternate vertices. Only even-sided polyhedra can be alternated, for example the zonohedra. Every 2n-sided face becomes n-sided...
, the Tetrahedron first forms the other triangular platonic solids, then duality makes the other two.
History
The Platonic solids have been known since antiquity. Ornamented models of them can be found among the
carved stone ballsCarved Stone Balls are petrospheres, usually round and rarely oval. They have from 3 to 160 protruding knobs on the surface. Their size is fairly uniform, they date from the late Neolithic to possibly the Iron Age and are mainly found in Scotland...
created by the late
neolithicThe Neolithic Age, Era, or Period, or New Stone Age, was a period in the development of human technology, beginning about 9500 BC in some parts of the Middle East, and later in other parts of the world. It is traditionally considered as the last part of the Stone Age...
people of
ScotlandScotland is a country that is part of the United Kingdom. Occupying the northern third of the island of Great Britain, it shares a border with England to the south and is bounded by the North Sea to the east, the Atlantic Ocean to the north and west, and the North Channel and Irish Sea to the...
at least 1000 years before Plato (Atiyah and Sutcliffe 2003). Dice go back to the dawn of civilization with shapes that augured formal charting of Platonic solids.
The ancient Greeks studied the Platonic solids extensively. Some sources (such as
ProclusProclus Lycaeus , called "The Successor" or "Diadochos" , was a Greek Neoplatonist philosopher, one of the last major Classical philosophers . He set forth one of the most elaborate and fully developed systems of Neoplatonism...
) credit
PythagorasPythagoras of Samos was an Ionian Greek philosopher, mathematician, and founder of the religious movement called Pythagoreanism. Most of the information about Pythagoras was written down centuries after he lived, so very little reliable information is known about him...
with their discovery. Other evidence suggests he may have only been familiar with the tetrahedron, cube, and dodecahedron, and that the discovery of the octahedron and icosahedron belong to
TheaetetusTheaetetus, Theaitētos, of Athens, possibly son of Euphronius, of the Athenian deme Sunium, was a classical Greek mathematician...
, a contemporary of Plato. In any case, Theaetetus gave a mathematical description of all five and may have been responsible for the first known proof that there are no other convex regular polyhedra.
The Platonic solids feature prominently in the philosophy of
PlatoPlato , was a Classical Greek philosopher, mathematician, student of Socrates, writer of philosophical dialogues, and founder of the Academy in Athens, the first institution of higher learning in the Western world. Along with his mentor, Socrates, and his student, Aristotle, Plato helped to lay the...
for whom they are named. Plato wrote about them in the dialogue
TimaeusTimaeus is one of Plato's dialogues, mostly in the form of a long monologue given by the title character, written circa 360 BC. The work puts forward speculation on the nature of the physical world and human beings. It is followed by the dialogue Critias.Speakers of the dialogue are Socrates,...
c.360 B.C. in which he associated each of the four
classical elementMany philosophies and worldviews have a set of classical elements believed to reflect the simplest essential parts and principles of which anything consists or upon which the constitution and fundamental powers of anything are based. Most frequently, classical elements refer to ancient beliefs...
s (
earthEarth, home and origin of humanity, has often been worshipped in its own right with its own unique spiritual tradition.-European tradition:Earth is one of the four classical elements in ancient Greek philosophy and science. It was commonly associated with qualities of heaviness, matter and the...
,
airAir is often seen as a universal power or pure substance. Its supposed fundamental importance to life can be seen in words such as aspire, inspire, perspire and spirit, all derived from the Latin spirare.-Greek and Roman tradition:...
,
waterWater is one of the elements in ancient Greek philosophy, in the Asian Indian system Panchamahabhuta, and in the Chinese cosmological and physiological system Wu Xing...
, and
fireFire has been an important part of all cultures and religions from pre-history to modern day and was vital to the development of civilization. It has been regarded in many different contexts throughout history, but especially as a metaphysical constant of the world.-Greek and Roman tradition:Fire...
) with a regular solid. Earth was associated with the cube, air with the octahedron, water with the icosahedron, and fire with the tetrahedron. There was intuitive justification for these associations: the heat of fire feels sharp and stabbing (like little tetrahedra). Air is made of the octahedron; its minuscule components are so smooth that one can barely feel it. Water, the icosahedron, flows out of one's hand when picked up, as if it is made of tiny little balls. By contrast, a highly un-spherical solid, the hexahedron (cube) represents earth. These clumsy little solids cause dirt to crumble and break when picked up, in stark difference to the smooth flow of water. Moreover, the solidity of the Earth was believed to be due to the fact that the cube is the only regular solid that tesselates
Euclidean spaceIn mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions...
. The fifth Platonic solid, the dodecahedron, Plato obscurely remarks, "...the god used for arranging the constellations on the whole heaven".
AristotleAristotle was a Greek philosopher and polymath, a student of Plato and teacher of Alexander the Great. His writings cover many subjects, including physics, metaphysics, poetry, theater, music, logic, rhetoric, linguistics, politics, government, ethics, biology, and zoology...
added a fifth element,
aithêrAccording to ancient and medieval science aether , also spelled æther or ether, is the material that fills the region of the universe above the terrestrial sphere.-Mythological origins:...
(aether in Latin, "ether" in English) and postulated that the heavens were made of this element, but he had no interest in matching it with Plato's fifth solid.
EuclidEuclid , fl. 300 BC, also known as Euclid of Alexandria, was a Greek mathematician, often referred to as the "Father of Geometry". He was active in Alexandria during the reign of Ptolemy I...
gave a complete mathematical description of the Platonic solids in the
ElementsEuclid's Elements is a mathematical and geometric treatise consisting of 13 books written by the Greek mathematician Euclid in Alexandria c. 300 BC. It is a collection of definitions, postulates , propositions , and mathematical proofs of the propositions...
, the last book (Book XIII) of which is devoted to their properties. Propositions 13–17 in Book XIII describe the construction of the tetrahedron, octahedron, cube, icosahedron, and dodecahedron in that order. For each solid Euclid finds the ratio of the diameter of the circumscribed sphere to the edge length. In Proposition 18 he argues that there are no further convex regular polyhedra.
Andreas SpeiserAndreas Speiser was a Swiss Mathematician and Philosopher of Science.-Life and work:Speiser studied since 1904 in Göttingen notably with David Hilbert, Felix Klein, Hermann Minkowski. In 1917 he became full time professor at the University of Zurich but later relocated in Basel...
has advocated the view that the construction of the 5 regular solids is the chief goal of the deductive system canonized in the
Elements. Much of the information in Book XIII is probably derived from the work of Theaetetus.
In the 16th century, the
GermanThe Germans are a Germanic ethnic group native to Central Europe. The English term Germans has referred to the German-speaking population of the Holy Roman Empire since the Late Middle Ages....
astronomerAn astronomer is a scientist who studies celestial bodies such as planets, stars and galaxies.Historically, astronomy was more concerned with the classification and description of phenomena in the sky, while astrophysics attempted to explain these phenomena and the differences between them using...
Johannes KeplerJohannes Kepler was a German mathematician, astronomer and astrologer. A key figure in the 17th century scientific revolution, he is best known for his eponymous laws of planetary motion, codified by later astronomers, based on his works Astronomia nova, Harmonices Mundi, and Epitome of Copernican...
attempted to find a relation between the five extraterrestrial
planetA planet is a celestial body orbiting a star or stellar remnant that is massive enough to be rounded by its own gravity, is not massive enough to cause thermonuclear fusion, and has cleared its neighbouring region of planetesimals.The term planet is ancient, with ties to history, science,...
s known at that time and the five Platonic solids. In
Mysterium CosmographicumMysterium Cosmographicum, is an astronomy book by the German astronomer Johannes Kepler, published at Tübingen in 1596 and in a second edition in 1621...
, published in 1596, Kepler laid out a model of the
solar systemThe Solar System consists of the Sun and the astronomical objects gravitationally bound in orbit around it, all of which formed from the collapse of a giant molecular cloud approximately 4.6 billion years ago. The vast majority of the system's mass is in the Sun...
in which the five solids were set inside one another and separated by a series of inscribed and circumscribed spheres. The six spheres each corresponded to one of the planets (
MercuryMercury is the innermost and smallest planet in the Solar System, orbiting the Sun once every 87.969 Earth days. The orbit of Mercury has the highest eccentricity of all the Solar System planets, and it has the smallest axial tilt. It completes three rotations about its axis for every two orbits...
,
VenusVenus is the second planet from the Sun, orbiting it every 224.7 Earth days. The planet is named after Venus, the Roman goddess of love and beauty. After the Moon, it is the brightest natural object in the night sky, reaching an apparent magnitude of −4.6, bright enough to cast shadows...
,
EarthEarth is the third planet from the Sun, and the densest and fifth-largest of the eight planets in the Solar System. It is also the largest of the Solar System's four terrestrial planets...
,
MarsMars is the fourth planet from the Sun in the Solar System. The planet is named after the Roman god of war, Mars. It is often described as the "Red Planet", as the iron oxide prevalent on its surface gives it a reddish appearance...
,
JupiterJupiter is the fifth planet from the Sun and the largest planet within the Solar System. It is a gas giant with mass one-thousandth that of the Sun but is two and a half times the mass of all the other planets in our Solar System combined. Jupiter is classified as a gas giant along with Saturn,...
, and
SaturnSaturn is the sixth planet from the Sun and the second largest planet in the Solar System, after Jupiter. Saturn is named after the Roman god Saturn, equated to the Greek Cronus , the Babylonian Ninurta and the Hindu Shani. Saturn's astronomical symbol represents the Roman god's sickle.Saturn,...
). The solids were ordered with the innermost being the octahedron, followed by the icosahedron, dodecahedron, tetrahedron, and finally the cube. In this way the structure of the solar system and the distance relationships between the planets was dictated by the Platonic solids. In the end, Kepler's original idea had to be abandoned, but out of his research came the recognition that the orbits of planets are ellipses rather than circles, as well as his two laws of orbital dynamics, changing the courses of physics and astronomy, plus the discovery of the Kepler solids.
Combinatorial properties
A convex polyhedron is a Platonic solid if and only if
- all its faces are congruent
In geometry, two figures are congruent if they have the same shape and size. This means that either object can be repositioned so as to coincide precisely with the other object...
convex regular polygonA regular polygon is a polygon that is equiangular and equilateral . Regular polygons may be convex or star.-General properties:...
s,
- none of its faces intersect except at their edges, and
- the same number of faces meet at each of its vertices
In geometry, a vertex is a special kind of point that describes the corners or intersections of geometric shapes.-Of an angle:...
.
Each Platonic solid can therefore be denoted by a symbol {
p,
q} where
- p = the number of edges of each face (or the number of vertices of each face) and
- q = the number of faces meeting at each vertex (or the number of edges meeting at each vertex).
The symbol {
p,
q}, called the
Schläfli symbol, gives a
combinatorialCombinatorics is a branch of mathematics concerning the study of finite or countable discrete structures. Aspects of combinatorics include counting the structures of a given kind and size , deciding when certain criteria can be met, and constructing and analyzing objects meeting the criteria ,...
description of the polyhedron. The Schläfli symbols of the five Platonic solids are given in the table below.
| Polyhedron |
Vertices |
Edges |
Faces |
Schläfli symbol |
Vertex
configurationIn geometry, a vertex configuration is a short-hand notation for representing the vertex figure of a polyhedron or tiling as the sequence of faces around a vertex. For uniform polyhedra there is only one vertex type and therefore the vertex configuration fully defines the polyhedron...
|
Historically corresponding element |
tetrahedronIn geometry, a tetrahedron is a polyhedron composed of four triangular faces, three of which meet at each vertex. A regular tetrahedron is one in which the four triangles are regular, or "equilateral", and is one of the Platonic solids...
|
|
4 |
6 |
4 |
{3, 3} |
3.3.3 |
Fire |
cubeIn geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. The cube can also be called a regular hexahedron and is one of the five Platonic solids. It is a special kind of square prism, of rectangular parallelepiped and...
/ hexahedronA hexahedron is any polyhedron with six faces, although usually implies the cube as a regular hexahedron with all its faces square, and three squares around each vertex....
|
|
8 |
12 |
6 |
{4, 3} |
4.4.4 |
Earth |
octahedronIn geometry, an octahedron is a polyhedron with eight faces. A regular octahedron is a Platonic solid composed of eight equilateral triangles, four of which meet at each vertex....
|
|
6 |
12 |
8 |
{3, 4} |
3.3.3.3 |
Air |
| dodecahedron |
|
20 |
30 |
12 |
{5, 3} |
5.5.5 |
Quintessence / Universe |
icosahedronIn geometry, an icosahedron is a regular polyhedron with 20 identical equilateral triangular faces, 30 edges and 12 vertices. It is one of the five Platonic solids....
|
|
12 |
30 |
20 |
{3, 5} |
3.3.3.3.3 |
Water |
All other combinatorial information about these solids, such as total number of vertices (
V), edges (
E), and faces (
F), can be determined from
p and
q. Since any edge joins two vertices and has two adjacent faces we must have:
The other relationship between these values is given by
Euler's formulaIn mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic is a topological invariant, a number that describes a topological space's shape or structure regardless of the way it is bent...
:
This nontrivial fact can be proved in a great variety of ways (in
algebraic topologyAlgebraic topology is a branch of mathematics which uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify up to homotopy equivalence.Although algebraic topology...
it follows from the fact that the Euler characteristic of the
sphereA sphere is a perfectly round geometrical object in three-dimensional space, such as the shape of a round ball. Like a circle in two dimensions, a perfect sphere is completely symmetrical around its center, with all points on the surface lying the same distance r from the center point...
is 2). Together these three relationships completely determine
V,
E, and
F:
Note that swapping
p and
q interchanges
F and
V while leaving
E unchanged (for a geometric interpretation of this fact, see the section on dual polyhedra below).
Classification
It is a classical result that there are only five convex regular polyhedra. Two common arguments are given below. Both of these arguments only show that there can be no more than five Platonic solids. That all five actually exist is a separate question—one that can be answered by an explicit construction.
Geometric proof
The following geometric argument is very similar to the one given by
EuclidEuclid , fl. 300 BC, also known as Euclid of Alexandria, was a Greek mathematician, often referred to as the "Father of Geometry". He was active in Alexandria during the reign of Ptolemy I...
in the
ElementsEuclid's Elements is a mathematical and geometric treatise consisting of 13 books written by the Greek mathematician Euclid in Alexandria c. 300 BC. It is a collection of definitions, postulates , propositions , and mathematical proofs of the propositions...
:
- Each vertex of the solid must coincide with one vertex each of at least three faces.
- At each vertex of the solid, the total, among the adjacent faces, of the angles between their respective adjacent sides must be less than 360°.
- The angles at all vertices of all faces of a Platonic solid are identical, so each vertex of each face must contribute less than 360°/3 = 120°.
- Regular polygons of six or more sides have only angles of 120° or more, so the common face must be the triangle, square, or pentagon. And for:
- Triangular
A triangle is one of the basic shapes of geometry: a polygon with three corners or vertices and three sides or edges which are line segments. A triangle with vertices A, B, and C is denoted ....
faces: each vertex of a regular triangle is 60°, so a shape may have 3, 4, or 5 triangles meeting at a vertex; these are the tetrahedron, octahedron, and icosahedron respectively.
- Square
In geometry, a square is a regular quadrilateral. This means that it has four equal sides and four equal angles...
faces: each vertex of a square is 90°, so there is only one arrangement possible with three faces at a vertex, the cube.
- Pentagon
In geometry, a pentagon is any five-sided polygon. A pentagon may be simple or self-intersecting. The sum of the internal angles in a simple pentagon is 540°. A pentagram is an example of a self-intersecting pentagon.- Regular pentagons :In a regular pentagon, all sides are equal in length and...
al faces: each vertex is 108°; again, only one arrangement, of three faces at a vertex is possible, the dodecahedron.
Topological proof
A purely
topologicalTopology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...
proof can be made using only combinatorial information about the solids. The key is
Euler's observationIn mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic is a topological invariant, a number that describes a topological space's shape or structure regardless of the way it is bent...
that
, and the fact that
, where
p stands for the number of edges of each face and
q for the number of edges meeting at each vertex. Combining these equations one obtains the equation
Simple algebraic manipulation then gives
Since
is strictly positive we must have
Using the fact that
p and
q must both be at least 3, one can easily see that there are only five possibilities for (
p,
q):
Angles
There are a number of
angleIn geometry, an angle is the figure formed by two rays sharing a common endpoint, called the vertex of the angle.Angles are usually presumed to be in a Euclidean plane with the circle taken for standard with regard to direction. In fact, an angle is frequently viewed as a measure of an circular arc...
s associated with each Platonic solid. The
dihedral angleIn geometry, a dihedral or torsion angle is the angle between two planes.The dihedral angle of two planes can be seen by looking at the planes "edge on", i.e., along their line of intersection...
is the interior angle between any two face planes. The dihedral angle, θ, of the solid {
p,
q} is given by the formula
This is sometimes more conveniently expressed in terms of the tangent by
The quantity
h is 4, 6, 6, 10, and 10 for the tetrahedron, cube, octahedron, dodecahedron, and icosahedron respectively.
The angular deficiency at the vertex of a polyhedron is the difference between the sum of the face-angles at that vertex and 2π. The defect, δ, at any vertex of the Platonic solids {
p,
q} is
By a theorem of Descartes, this is equal to 4π divided by the number of vertices (i.e. the total defect at all vertices is 4π).
The 3-dimensional analog of a plane angle is a
solid angleThe solid angle, Ω, is the two-dimensional angle in three-dimensional space that an object subtends at a point. It is a measure of how large that object appears to an observer looking from that point...
. The solid angle, Ω, at the vertex of a Platonic solid is given in terms of the dihedral angle by
This follows from the spherical excess formula for a spherical polygon and the fact that the
vertex figureIn geometry a vertex figure is, broadly speaking, the figure exposed when a corner of a polyhedron or polytope is sliced off.-Definitions - theme and variations:...
of the polyhedron {
p,
q} is a regular
q-gon.
The solid angle of a face subtended from the center of a platonic solid is equal to the solid angle of a full sphere (4π steradians) divided by the number of faces. Note that this is equal to the angular deficiency of its dual.
The various angles associated with the Platonic solids are tabulated below. The numerical values of the solid angles are given in
steradianThe steradian is the SI unit of solid angle. It is used to describe two-dimensional angular spans in three-dimensional space, analogous to the way in which the radian describes angles in a plane...
s. The constant φ = (1+√5)/2 is the
golden ratioIn mathematics and the arts, two quantities are in the golden ratio if the ratio of the sum of the quantities to the larger quantity is equal to the ratio of the larger quantity to the smaller one. The golden ratio is an irrational mathematical constant, approximately 1.61803398874989...
.
| Polyhedron |
Dihedral angleIn geometry, a dihedral or torsion angle is the angle between two planes.The dihedral angle of two planes can be seen by looking at the planes "edge on", i.e., along their line of intersection...
 |
 |
Vertex angleIn geometry, a vertex angle is the angle associated with a vertex of a polygon.The vertex angle in a polygon is measured by the interior side of the vertex. For any simple n-gon, the sum of the interior angles is π radians or 180 degrees....
|
DefectIn geometry, the defect means the failure of some angles to add up to the expected amount of 360° or 180°, when such angles in the plane would...
( ) |
Vertex solid angleThe solid angle, Ω, is the two-dimensional angle in three-dimensional space that an object subtends at a point. It is a measure of how large that object appears to an observer looking from that point...
( ) |
Face
solid angle |
tetrahedronIn geometry, a tetrahedron is a polyhedron composed of four triangular faces, three of which meet at each vertex. A regular tetrahedron is one in which the four triangles are regular, or "equilateral", and is one of the Platonic solids...
|
70.53° |
 |
60° |
 |
 |
 |
 |
cubeIn geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. The cube can also be called a regular hexahedron and is one of the five Platonic solids. It is a special kind of square prism, of rectangular parallelepiped and...
|
90° |
 |
90° |
 |
 |
 |
 |
octahedronIn geometry, an octahedron is a polyhedron with eight faces. A regular octahedron is a Platonic solid composed of eight equilateral triangles, four of which meet at each vertex....
|
109.47° |
 |
60°, 90° |
 |
 |
 |
 |
| dodecahedron |
116.57° |
 |
108° |
 |
 |
 |
 |
icosahedronIn geometry, an icosahedron is a regular polyhedron with 20 identical equilateral triangular faces, 30 edges and 12 vertices. It is one of the five Platonic solids....
|
138.19° |
 |
60°, 108° |
 |
 |
 |
 |
Radii, area, and volume
Another virtue of regularity is that the Platonic solids all possess three concentric spheres:
- the circumscribed sphere
In geometry, a circumscribed sphere of a polyhedron is a sphere that contains the polyhedron and touches each of the polyhedron's vertices. The word circumsphere is sometimes used to mean the same thing...
which passes through all the vertices,
- the midsphere
In geometry, the midsphere or intersphere of a polyhedron is a sphere which is tangent to every edge of the polyhedron. That is to say, it touches any given edge at exactly one point...
which is tangent to each edge at the midpoint of the edge, and
- the inscribed sphere
In geometry, the inscribed sphere or insphere of a convex polyhedron is a sphere that is contained within the polyhedron and tangent to each of the polyhedron's faces...
which is tangent to each face at the center of the face.
The
radiiIn classical geometry, a radius of a circle or sphere is any line segment from its center to its perimeter. By extension, the radius of a circle or sphere is the length of any such segment, which is half the diameter. If the object does not have an obvious center, the term may refer to its...
of these spheres are called the
circumradius, the
midradius, and the
inradius. These are the distances from the center of the polyhedron to the vertices, edge midpoints, and face centers respectively. The circumradius
R and the inradius
r of the solid {
p,
q} with edge length
a are given by
where θ is the dihedral angle. The midradius ρ is given by
where
h is the quantity used above in the definition of the dihedral angle (
h = 4, 6, 6, 10, or 10). Note that the ratio of the circumradius to the inradius is symmetric in
p and
q:
The
surface areaSurface area is the measure of how much exposed area a solid object has, expressed in square units. Mathematical description of the surface area is considerably more involved than the definition of arc length of a curve. For polyhedra the surface area is the sum of the areas of its faces...
,
A, of a Platonic solid {
p,
q} is easily computed as area of a regular
p-gon times the number of faces
F. This is:
The
volumeVolume is the quantity of three-dimensional space enclosed by some closed boundary, for example, the space that a substance or shape occupies or contains....
is computed as
F times the volume of the
pyramidIn geometry, a pyramid is a polyhedron formed by connecting a polygonal base and a point, called the apex. Each base edge and apex form a triangle. It is a conic solid with polygonal base....
whose base is a regular
p-gon and whose height is the inradius
r. That is,
The following table lists the various radii of the Platonic solids together with their surface area and volume. The overall size is fixed by taking the edge length,
a, to be equal to 2.
Polyhedron
(a = 2) |
Inradius (r) |
Midradius (ρ) |
Circumradius (R) |
Surface area (A) |
Volume (V) |
tetrahedronIn geometry, a tetrahedron is a polyhedron composed of four triangular faces, three of which meet at each vertex. A regular tetrahedron is one in which the four triangles are regular, or "equilateral", and is one of the Platonic solids...
|
 |
 |
 |
 |
 |
cubeIn geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. The cube can also be called a regular hexahedron and is one of the five Platonic solids. It is a special kind of square prism, of rectangular parallelepiped and...
|
 |
 |
 |
 |
 |
octahedronIn geometry, an octahedron is a polyhedron with eight faces. A regular octahedron is a Platonic solid composed of eight equilateral triangles, four of which meet at each vertex....
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| dodecahedron |
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icosahedronIn geometry, an icosahedron is a regular polyhedron with 20 identical equilateral triangular faces, 30 edges and 12 vertices. It is one of the five Platonic solids....
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The constants φ and ξ in the above are given by
Among the Platonic solids, either the dodecahedron or the icosahedron may be seen as the best approximation to the sphere. The icosahedron has the largest number of faces and the largest dihedral angle, it hugs its inscribed sphere the tightest, and its surface area to volume ratio is closest to that of a sphere of the same size (i.e. either the same surface area or the same volume.) The dodecahedron, on the other hand, has the smallest angular defect, the largest vertex solid angle, and it fills out its circumscribed sphere the most.
Dual polyhedra
Every polyhedron has a
dual (or "polar") polyhedronIn geometry, polyhedra are associated into pairs called duals, where the vertices of one correspond to the faces of the other. The dual of the dual is the original polyhedron. The dual of a polyhedron with equivalent vertices is one with equivalent faces, and of one with equivalent edges is another...
with faces and vertices interchanged. The dual of every Platonic solid is another Platonic solid, so that we can arrange the five solids into dual pairs.
- The tetrahedron is self-dual (i.e. its dual is another tetrahedron).
- The cube and the octahedron form a dual pair.
- The dodecahedron and the icosahedron form a dual pair.
If a polyhedron has Schläfli symbol {
p,
q}, then its dual has the symbol {
q,
p}. Indeed every combinatorial property of one Platonic solid can be interpreted as another combinatorial property of the dual.
One can construct the dual polyhedron by taking the vertices of the dual to be the centers of the faces of the original figure. The edges of the dual are formed by connecting the centers of adjacent faces in the original. In this way, the number of faces and vertices is interchanged, while the number of edges stays the same.
More generally, one can dualize a Platonic solid with respect to a sphere of radius
d concentric with the solid. The radii (
R, ρ,
r) of a solid and those of its dual (
R*, ρ*,
r*) are related by
It is often convenient to dualize with respect to the midsphere (
d = ρ) since it has the same relationship to both polyhedra. Taking
d2 =
Rr gives a dual solid with the same circumradius and inradius (i.e.
R* =
R and
r* =
r).
Symmetry groups
In mathematics, the concept of
symmetrySymmetry generally conveys two primary meanings. The first is an imprecise sense of harmonious or aesthetically pleasing proportionality and balance; such that it reflects beauty or perfection...
is studied with the notion of a
mathematical groupIn mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...
. Every polyhedron has an associated
symmetry groupThe symmetry group of an object is the group of all isometries under which it is invariant with composition as the operation...
, which is the set of all transformations (Euclidean isometries) which leave the polyhedron invariant. The
orderIn group theory, a branch of mathematics, the term order is used in two closely related senses:* The order of a group is its cardinality, i.e., the number of its elements....
of the symmetry group is the number of symmetries of the polyhedron. One often distinguishes between the
full symmetry group, which includes
reflectionIn mathematics, a reflection is a mapping from a Euclidean space to itself that is an isometry with a hyperplane as set of fixed points; this set is called the axis or plane of reflection. The image of a figure by a reflection is its mirror image in the axis or plane of reflection...
s, and the
proper symmetry group, which includes only
rotationIn geometry and linear algebra, a rotation is a transformation in a plane or in space that describes the motion of a rigid body around a fixed point. A rotation is different from a translation, which has no fixed points, and from a reflection, which "flips" the bodies it is transforming...
s.
The symmetry groups of the Platonic solids are known as
polyhedral groupIn geometry, the polyhedral group is any of the symmetry groups of the Platonic solids. There are three polyhedral groups:*The tetrahedral group of order 12, rotational symmetry group of the regular tetrahedron....
s (which are a special class of the
point groups in three dimensionsIn geometry, a point group in three dimensions is an isometry group in three dimensions that leaves the origin fixed, or correspondingly, an isometry group of a sphere. It is a subgroup of the orthogonal group O, the group of all isometries that leave the origin fixed, or correspondingly, the group...
). The high degree of symmetry of the Platonic solids can be interpreted in a number of ways. Most importantly, the vertices of each solid are all equivalent under the
actionIn algebra and geometry, a group action is a way of describing symmetries of objects using groups. The essential elements of the object are described by a set, and the symmetries of the object are described by the symmetry group of this set, which consists of bijective transformations of the set...
of the symmetry group, as are the edges and faces. One says the action of the symmetry group is transitive on the vertices, edges, and faces. In fact, this is another way of defining regularity of a polyhedron: a polyhedron is
regular if and only if it is vertex-uniform,
edge-uniformIn geometry, a polytope is isotoxal or edge-transitive if its symmetries act transitively on its edges...
, and
face-uniformIn geometry, a polytope or tiling is isohedral or face-transitive when all its faces are the same. More specifically, all faces must be not merely congruent but must be transitive, i.e. must lie within the same symmetry orbit.Isohedral polyhedra are called isohedra. They can be described by their...
.
There are only three symmetry groups associated with the Platonic solids rather than five, since the symmetry group of any polyhedron coincides with that of its dual. This is easily seen by examining the construction of the dual polyhedron. Any symmetry of the original must be a symmetry of the dual and vice-versa. The three polyhedral groups are:
- the tetrahedral group T,
- the octahedral group O (which is also the symmetry group of the cube), and
- the icosahedral group I (which is also the symmetry group of the dodecahedron).
The orders of the proper (rotation) groups are 12, 24, and 60 respectively — precisely twice the number of edges in the respective polyhedra. The orders of the full symmetry groups are twice as much again (24, 48, and 120). See (Coxeter 1973) for a derivation of these facts. All Platonic solids except the tetrahedron are
centrally symmetric, meaning they are preserved under reflection through the origin.
The following table lists the various symmetry properties of the Platonic solids. The symmetry groups listed are the full groups with the rotation subgroups given in parenthesis (likewise for the number of symmetries). Wythoff's kaleidoscope construction is a method for constructing polyhedra directly from their symmetry groups. We list for reference Wythoff's symbol for each of the Platonic solids.
| Polyhedron |
Schläfli symbol |
Wythoff symbol In geometry, the Wythoff symbol was first used by Coxeter, Longeut-Higgens and Miller in their enumeration of the uniform polyhedra. It represents a construction by way of Wythoff's construction applied to Schwarz triangles....
|
Dual polyhedron In geometry, polyhedra are associated into pairs called duals, where the vertices of one correspond to the faces of the other. The dual of the dual is the original polyhedron. The dual of a polyhedron with equivalent vertices is one with equivalent faces, and of one with equivalent edges is another...
|
Symmetries |
Symmetry groupThe symmetry group of an object is the group of all isometries under which it is invariant with composition as the operation...
|
tetrahedronIn geometry, a tetrahedron is a polyhedron composed of four triangular faces, three of which meet at each vertex. A regular tetrahedron is one in which the four triangles are regular, or "equilateral", and is one of the Platonic solids...
|
{3, 3} |
3 > 2 3 |
tetrahedron |
24 (12) |
Td (T) 150px|right|thumb|A regular [[tetrahedron]], an example of a solid with full tetrahedral symmetryA regular tetrahedron has 12 rotational symmetries, and a symmetry order of 24 including transformations that combine a reflection and a rotation.The group of all symmetries is isomorphic to the group...
|
cubeIn geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. The cube can also be called a regular hexahedron and is one of the five Platonic solids. It is a special kind of square prism, of rectangular parallelepiped and...
|
{4, 3} |
3 > 2 4 |
octahedron |
48 (24) |
Oh (O) 150px|thumb|right|The [[cube]] is the most common shape with octahedral symmetryA regular octahedron has 24 rotational symmetries, and a symmetry order of 48 including transformations that combine a reflection and a rotation...
|
octahedronIn geometry, an octahedron is a polyhedron with eight faces. A regular octahedron is a Platonic solid composed of eight equilateral triangles, four of which meet at each vertex....
|
{3, 4} |
4 > 2 3 |
cube |
| dodecahedron |
{5, 3} |
3 > 2 5 |
icosahedron |
120 (60) |
Ih (I) A regular icosahedron has 60 rotational symmetries, and a symmetry order of 120 including transformations that combine a reflection and a rotation...
|
icosahedronIn geometry, an icosahedron is a regular polyhedron with 20 identical equilateral triangular faces, 30 edges and 12 vertices. It is one of the five Platonic solids....
|
{3, 5} |
5 > 2 3 |
dodecahedron |
In nature and technology
The tetrahedron, cube, and octahedron all occur naturally in
crystal structureIn mineralogy and crystallography, crystal structure is a unique arrangement of atoms or molecules in a crystalline liquid or solid. A crystal structure is composed of a pattern, a set of atoms arranged in a particular way, and a lattice exhibiting long-range order and symmetry...
s. These by no means exhaust the numbers of possible forms of crystals. However, neither the regular icosahedron nor the regular dodecahedron are amongst them. One of the forms, called the
pyritohedronIn geometry, a pyritohedron is an irregular dodecahedron with pyritohedral symmetry. Like the regular dodecahedron, it has twelve identical pentagonal faces, with three meeting in each of the 20 vertices. However, the pentagons are not regular, and the structure has no fivefold symmetry axes...
(named for the group of
mineralsThe mineral pyrite, or iron pyrite, is an iron sulfide with the formula FeS2. This mineral's metallic luster and pale-to-normal, brass-yellow hue have earned it the nickname fool's gold because of its resemblance to gold...
of which it is typical) has twelve pentagonal faces, arranged in the same pattern as the faces of the regular dodecahedron. The faces of the pyritohedron are, however, not regular, so the pyritohedron is also not regular.
In the early 20th century,
Ernst HaeckelThe "European War" became known as "The Great War", and it was not until 1920, in the book "The First World War 1914-1918" by Charles à Court Repington, that the term "First World War" was used as the official name for the conflict.-Research:...
described (Haeckel, 1904) a number of species of Radiolaria, some of whose skeletons are shaped like various regular polyhedra. Examples include
Circoporus octahedrus,
Circogonia icosahedra,
Lithocubus geometricus and
Circorrhegma dodecahedra. The shapes of these creatures should be obvious from their names.
Many
virusA virus is a small infectious agent that can replicate only inside the living cells of organisms. Viruses infect all types of organisms, from animals and plants to bacteria and archaea...
es, such as the herpes virus, have the shape of a regular icosahedron. Viral structures are built of repeated identical
proteinProteins are biochemical compounds consisting of one or more polypeptides typically folded into a globular or fibrous form, facilitating a biological function. A polypeptide is a single linear polymer chain of amino acids bonded together by peptide bonds between the carboxyl and amino groups of...
subunits and the icosahedron is the easiest shape to assemble using these subunits. A regular polyhedron is used because it can be built from a single basic unit protein used over and over again; this saves space in the viral
genomeIn modern molecular biology and genetics, the genome is the entirety of an organism's hereditary information. It is encoded either in DNA or, for many types of virus, in RNA. The genome includes both the genes and the non-coding sequences of the DNA/RNA....
.
In
meteorologyMeteorology is the interdisciplinary scientific study of the atmosphere. Studies in the field stretch back millennia, though significant progress in meteorology did not occur until the 18th century. The 19th century saw breakthroughs occur after observing networks developed across several countries...
and
climatologyClimatology is the study of climate, scientifically defined as weather conditions averaged over a period of time, and is a branch of the atmospheric sciences...
, global numerical models of atmospheric flow are of increasing interest which employ grids that are based on an icosahedron (refined by
triangulationIn trigonometry and geometry, triangulation is the process of determining the location of a point by measuring angles to it from known points at either end of a fixed baseline, rather than measuring distances to the point directly...
) instead of the more commonly used
longitudeLongitude is a geographic coordinate that specifies the east-west position of a point on the Earth's surface. It is an angular measurement, usually expressed in degrees, minutes and seconds, and denoted by the Greek letter lambda ....
/
latitudeIn geography, the latitude of a location on the Earth is the angular distance of that location south or north of the Equator. The latitude is an angle, and is usually measured in degrees . The equator has a latitude of 0°, the North pole has a latitude of 90° north , and the South pole has a...
grid. This has the advantage of evenly distributed spatial resolution without
singularitiesIn mathematics, a singularity is in general a point at which a given mathematical object is not defined, or a point of an exceptional set where it fails to be well-behaved in some particular way, such as differentiability...
(i.e. the
polesthumb|right|180px|The state flag of [[Poland]] as used by Polish government and diplomatic authoritiesThe Polish people, or Poles , are a nation indigenous to Poland. They are united by the Polish language, which belongs to the historical Lechitic subgroup of West Slavic languages of Central Europe...
) at the expense of somewhat greater numerical difficulty.
Geometry of
space frameA space frame or space structure is a truss-like, lightweight rigid structure constructed from interlocking struts in a geometric pattern. Space frames can be used to span large areas with few interior supports...
s is often based on platonic solids. In MERO system, Platonic solids are used for naming convention of various space frame configurations. For example ½O+T refers to a configuration made of one half of octahedron and a tetrahedron.
Several
Platonic hydrocarbonsPlatonic hydrocarbons are the molecular representation of platonic solid geometries with vertices replaced by carbon atoms and with edges replaced by chemical bonds...
have been synthesised, including
cubaneCubane is a synthetic hydrocarbon molecule that consists of eight carbon atoms arranged at the corners of a cube, with one hydrogen atom attached to each carbon atom. A solid crystalline substance, cubane is one of the Platonic hydrocarbons. It was first synthesized in 1964 by Philip Eaton, a...
and
dodecahedraneDodecahedrane is a chemical compound first synthesised by Leo Paquette of Ohio State University in 1982, primarily for the "aesthetically pleasing symmetry of the dodecahedral framework"....
.
Platonic solids are often used to make
diceA die is a small throwable object with multiple resting positions, used for generating random numbers...
, because dice of these shapes can be made fair. 6-sided dice are very common, but the other numbers are commonly used in
role-playing gameA role-playing game is a game in which players assume the roles of characters in a fictional setting. Players take responsibility for acting out these roles within a narrative, either through literal acting, or through a process of structured decision-making or character development...
s. Such dice are commonly referred to as d
n where
n is the number of faces (d8, d20, etc.); see
dice notationDice notation is a system to represent different combinations of dice in role-playing games using simple algebra-like notation such as 2d6+12....
for more details.
These shapes frequently show up in other games or puzzles. Puzzles similar to a
Rubik's CubeRubik's Cube is a 3-D mechanical puzzle invented in 1974 by Hungarian sculptor and professor of architecture Ernő Rubik.Originally called the "Magic Cube", the puzzle was licensed by Rubik to be sold by Ideal Toy Corp. in 1980 and won the German Game of the Year special award for Best Puzzle that...
come in all five shapes — see magic polyhedra.
Liquid Crystals with symmetries of Platonic Solids
For the intermediate material phase called Liquid Crystals
the existence of such symmetries was first proposed in 1981
by
H. KleinertHagen Kleinert is Professor of Theoretical Physics at the Free University of Berlin, Germany , at theWest University of Timişoara, at thein Bishkek. He is also of the...
and K. Maki and their structure was analyzed in. See the review article
here.
In aluminum the icosahedral structure was discovered three years after this
by
Dan ShechtmanDan Shechtman is the Philip Tobias Professor of Materials Science at the Technion – Israel Institute of Technology, an Associate of the US Department of Energy's Ames Laboratory, and Professor of Materials Science at Iowa State University. On April 8, 1982, while on sabbatical at the U.S...
, which earned him the Nobel Prize in 2011.
Uniform polyhedra
There exist four regular polyhedra which are not convex, called Kepler–Poinsot polyhedra. These all have
icosahedral symmetryA regular icosahedron has 60 rotational symmetries, and a symmetry order of 120 including transformations that combine a reflection and a rotation...
and may be obtained as
stellationStellation is a process of constructing new polygons , new polyhedra in three dimensions, or, in general, new polytopes in n dimensions. The process consists of extending elements such as edges or face planes, usually in a symmetrical way, until they meet each other again...
s of the dodecahedron and the icosahedron.
cuboctahedronIn geometry, a cuboctahedron is a polyhedron with eight triangular faces and six square faces. A cuboctahedron has 12 identical vertices, with two triangles and two squares meeting at each, and 24 identical edges, each separating a triangle from a square. As such it is a quasiregular polyhedron,...
|
icosidodecahedronIn geometry, an icosidodecahedron is a polyhedron with twenty triangular faces and twelve pentagonal faces. An icosidodecahedron has 30 identical vertices, with two triangles and two pentagons meeting at each, and 60 identical edges, each separating a triangle from a pentagon...
|
The next most regular convex polyhedra after the Platonic solids are the
cuboctahedronIn geometry, a cuboctahedron is a polyhedron with eight triangular faces and six square faces. A cuboctahedron has 12 identical vertices, with two triangles and two squares meeting at each, and 24 identical edges, each separating a triangle from a square. As such it is a quasiregular polyhedron,...
, which is a
rectificationIn Euclidean geometry, rectification is the process of truncating a polytope by marking the midpoints of all its edges, and cutting off its vertices at those points...
of the cube and the octahedron, and the
icosidodecahedronIn geometry, an icosidodecahedron is a polyhedron with twenty triangular faces and twelve pentagonal faces. An icosidodecahedron has 30 identical vertices, with two triangles and two pentagons meeting at each, and 60 identical edges, each separating a triangle from a pentagon...
, which is a rectification of the dodecahedron and the icosahedron (the rectification of the self-dual tetrahedron is a regular octahedron). These are both
quasi-regular, meaning that they are vertex- and edge-uniform and have regular faces, but the faces are not all congruent (coming in two different classes). They form two of the thirteen
Archimedean solidIn geometry an Archimedean solid is a highly symmetric, semi-regular convex polyhedron composed of two or more types of regular polygons meeting in identical vertices...
s, which are the convex
uniform polyhedraA uniform polyhedron is a polyhedron which has regular polygons as faces and is vertex-transitive...
with polyhedral symmetry.
The uniform polyhedra form a much broader class of polyhedra. These figures are vertex-uniform and have one or more types of
regularA regular polygon is a polygon that is equiangular and equilateral . Regular polygons may be convex or star.-General properties:...
or
star polygons for faces. These include all the polyhedra mentioned above together with an infinite set of
prismIn geometry, a prism is a polyhedron with an n-sided polygonal base, a translated copy , and n other faces joining corresponding sides of the two bases. All cross-sections parallel to the base faces are the same. Prisms are named for their base, so a prism with a pentagonal base is called a...
s, an infinite set of
antiprismIn geometry, an n-sided antiprism is a polyhedron composed of two parallel copies of some particular n-sided polygon, connected by an alternating band of triangles...
s, and 53 other non-convex forms.
The
Johnson solidIn geometry, a Johnson solid is a strictly convex polyhedron, each face of which is a regular polygon, but which is not uniform, i.e., not a Platonic solid, Archimedean solid, prism or antiprism. There is no requirement that each face must be the same polygon, or that the same polygons join around...
s are convex polyhedra which have regular faces but are not uniform.
Tessellations
The three regular tessellations of the plane are closely related to the Platonic solids. Indeed, one can view the Platonic solids as the five regular tessellations of the
sphereA sphere is a perfectly round geometrical object in three-dimensional space, such as the shape of a round ball. Like a circle in two dimensions, a perfect sphere is completely symmetrical around its center, with all points on the surface lying the same distance r from the center point...
. This is done by projecting each solid onto a concentric sphere. The faces project onto regular spherical polygons which exactly cover the sphere. One can show that every regular tessellation of the sphere is characterized by a pair of integers {
p,
q} with 1/
p + 1/
q > 1/2. Likewise, a regular tessellation of the plane is characterized by the condition 1/
p + 1/
q = 1/2. There are three possibilities:
In a similar manner one can consider regular tessellations of the
hyperbolic planeIn mathematics, hyperbolic geometry is a non-Euclidean geometry, meaning that the parallel postulate of Euclidean geometry is replaced...
. These are characterized by the condition 1/
p + 1/
q < 1/2. There is an infinite family of such tessellations.
Higher dimensions
In more than three dimensions, polyhedra generalize to
polytopeIn elementary geometry, a polytope is a geometric object with flat sides, which exists in any general number of dimensions. A polygon is a polytope in two dimensions, a polyhedron in three dimensions, and so on in higher dimensions...
s, with higher-dimensional convex
regular polytopeIn mathematics, a regular polytope is a polytope whose symmetry is transitive on its flags, thus giving it the highest degree of symmetry. All its elements or j-faces — cells, faces and so on — are also transitive on the symmetries of the polytope, and are regular polytopes of...
s being the equivalents of the three-dimensional Platonic solids.
In the mid-19th century the Swiss mathematician
Ludwig SchläfliLudwig Schläfli was a Swiss geometer and complex analyst who was one of the key figures in developing the notion of higher dimensional spaces. The concept of multidimensionality has since come to play a pivotal role in physics, and is a common element in science fiction...
discovered the four-dimensional analogues of the Platonic solids, called
convex regular 4-polytopeIn mathematics, a convex regular 4-polytope is a 4-dimensional polytope that is both regular and convex. These are the four-dimensional analogs of the Platonic solids and the regular polygons ....
s. There are exactly six of these figures; five are analogous to the Platonic solids, while the sixth one, the
24-cell, has one lower-dimension analogue (Truncation of a simplex-faceted polyhedron that has simplices for ridges and is self-dual): the
Hexagon.
In all dimensions higher than four, there are only three convex regular polytopes: the
simplexIn geometry, a simplex is a generalization of the notion of a triangle or tetrahedron to arbitrary dimension. Specifically, an n-simplex is an n-dimensional polytope which is the convex hull of its n + 1 vertices. For example, a 2-simplex is a triangle, a 3-simplex is a tetrahedron,...
, the
hypercubeIn geometry, a hypercube is an n-dimensional analogue of a square and a cube . It is a closed, compact, convex figure whose 1-skeleton consists of groups of opposite parallel line segments aligned in each of the space's dimensions, perpendicular to each other and of the same length.An...
, and the
cross-polytopeIn geometry, a cross-polytope, orthoplex, hyperoctahedron, or cocube is a regular, convex polytope that exists in any number of dimensions. The vertices of a cross-polytope are all the permutations of . The cross-polytope is the convex hull of its vertices...
. In three dimensions, these coincide with the tetrahedron, the cube, and the octahedron.
See also
- Archimedean solid
In geometry an Archimedean solid is a highly symmetric, semi-regular convex polyhedron composed of two or more types of regular polygons meeting in identical vertices...
- Catalan solid
In mathematics, a Catalan solid, or Archimedean dual, is a dual polyhedron to an Archimedean solid. The Catalan solids are named for the Belgian mathematician, Eugène Catalan, who first described them in 1865....
- Kepler solids
- List of regular polytopes
- Project Euler
Project Euler is a website dedicated to a series of computational problems intended to be solved with computer programs. The project attracts adults and students interested in mathematics and computer programming. , it includes 351 problems of varying difficulty, each solvable in less than a...
uses platonic solids to denote scoring levels.
- Regular polytope
In mathematics, a regular polytope is a polytope whose symmetry is transitive on its flags, thus giving it the highest degree of symmetry. All its elements or j-faces — cells, faces and so on — are also transitive on the symmetries of the polytope, and are regular polytopes of...
s
- Metatron's Cube
Metatron's Cube is a two-dimensional geometric figure created from 13 equal circles with lines from the center of each circle extending out to the centers of the other 12 circles. Six circles are placed in a hexagonal pattern around a central circle, with six more extending out along the same...
External links