Platonic solid

In geometry, a Platonic solid is defined as a convex, regular polyhedron situated within three-dimensional Euclidean space. The characteristic of being a regular polyhedron implies that its faces are congruent regular polygons, meaning they are identical in shape and size, possess congruent angles, and have congruent edges. Furthermore, an equal number of faces converge at every vertex. Only five such polyhedra exist: the tetrahedron, featuring four triangular faces; the cube, with six square faces; the octahedron, comprising eight triangular faces; the dodecahedron, distinguished by twelve pentagonal faces; and the icosahedron, which has twenty triangular faces.

The study of Platonic solids has engaged geometers for millennia. These geometric figures derive their name from the ancient Greek philosopher Plato, who, in his dialogue Timaeus, posited that the fundamental classical elements were constituted by these regular solids.

History

The existence of Platonic solids has been recognized since antiquity. Speculation suggests that certain carved stone balls, produced by late Neolithic communities in Scotland, might represent these forms. Nevertheless, these artifacts feature rounded knobs instead of exhibiting polyhedral structures. The quantity of knobs often diverged from the vertex counts of the Platonic solids. Specifically, no ball has been found with knobs corresponding to the 20 vertices of a dodecahedron, and the configuration of these knobs was not consistently symmetrical.

Extensive research into Platonic solids was conducted by the ancient Greeks. While some historical accounts, such as those by Proclus, attribute their discovery to Pythagoras, alternative evidence indicates that Pythagoras might have been acquainted solely with the tetrahedron, cube, and dodecahedron. The discovery of the octahedron and icosahedron is instead often credited to Theaetetus, a contemporary of Plato. Regardless, Theaetetus provided a comprehensive mathematical description of all five solids and is potentially responsible for the earliest known proof demonstrating the non-existence of any other convex regular polyhedra.

The Platonic solids hold a significant position within the philosophy of Plato, from whom they derive their name. Plato discussed these forms in his dialogue Timaeus, composed c. 360 B.C., wherein he correlated each of the four classical elements—earth, air, water, and fire—with a specific regular solid. Earth was linked to the cube, air to the octahedron, water to the icosahedron, and fire to the tetrahedron. Regarding the fifth Platonic solid, the dodecahedron, Plato made an enigmatic statement, suggesting that "...the god used [it] for arranging the constellations on the whole heaven." Subsequently, Aristotle introduced a fifth element, aither (known as aether in Latin and "ether" in English), postulating that the celestial spheres were composed of this substance, though he did not endeavor to align it with Plato's fifth solid.

Euclid provided a comprehensive mathematical description of the Platonic solids in his seminal work, the Elements, with the final volume (Book XIII) dedicated entirely to their properties. Propositions 13–17 within Book XIII detail the construction sequence for the tetrahedron, octahedron, cube, icosahedron, and dodecahedron. For each solid, Euclid determined the ratio between the diameter of its circumscribed sphere and its edge length. In Proposition 18, he asserts that no additional convex regular polyhedra exist. Andreas Speiser has advanced the perspective that the construction of these five regular solids represents the primary objective of the deductive system formalized in the Elements. A substantial portion of the content found in Book XIII is likely derived from Theaetetus's earlier work.

In the 16th century, the German astronomer Johannes Kepler endeavored to establish a relationship between the five extraterrestrial planets known at that time and the five Platonic solids. In his work Mysterium Cosmographicum, published in 1596, Kepler proposed a Solar System model where the five solids were nested within one another, separated by a sequence of inscribed and circumscribed spheres. Kepler theorized that the distance relationships among the six planets then identified could be understood through the arrangement of the five Platonic solids enclosed within a sphere representing Saturn's orbit. Each of the six spheres corresponded to one planet (Mercury, Venus, Earth, Mars, Jupiter, and Saturn). The solids were ordered from innermost to outermost as the octahedron, followed by the icosahedron, dodecahedron, tetrahedron, and finally the cube, thereby positing that Platonic solids dictated the Solar System's structure and planetary distances. Ultimately, Kepler's initial hypothesis proved untenable and was abandoned; however, his research led to the formulation of his three laws of orbital dynamics. The first of these laws, stating that planetary orbits are ellipses rather than circles, fundamentally altered the trajectory of physics and astronomy. He also identified the Kepler solids, which comprise two nonconvex regular polyhedra.

Cartesian coordinates

For Platonic solids centered at the origin, the simple Cartesian coordinates of their vertices are presented below. The Greek letter ${\displaystyle \varphi }$ is utilized to denote the golden ratio, which is defined as §2627§ + §3233§ §3738§ ≈ 1.6180 {\displaystyle {\frac {1+{\sqrt {5}}}{2}}\approx 1.6180} .

The coordinates for the tetrahedron, dodecahedron, and icosahedron are provided in two configurations, each derivable from the other. For the tetrahedron, this derivation involves altering the sign of all coordinates, representing central symmetry. In the other cases, it entails exchanging two coordinates, corresponding to a reflection across any of the three diagonal planes.

These coordinates reveal specific relationships among the Platonic solids. The vertices of a tetrahedron, for instance, constitute half of those of a cube, denoted as {4,3} or . This corresponds to one of two sets of four vertices in dual positions, represented as h{4,3} or . The combination of both tetrahedral positions forms the compound stellated octahedron.

The coordinates of the icosahedron are related to two alternating sets of coordinates derived from a nonuniform truncated octahedron, designated t{3,4} or . This figure is also known as a snub octahedron, represented as s{3,4} or , and its relationship is evident in the compound of two icosahedra.

Eight vertices of the dodecahedron are coincident with those of a cube. When all possible orientations are considered, this relationship culminates in the formation of the compound of five cubes.

Combinatorial properties

A convex polyhedron qualifies as a Platonic solid exclusively when all three of the subsequent criteria are satisfied.

• All faces must be congruent, convex, and regular polygons.
• Faces may not intersect each other except along their shared edges.
• An identical number of faces must converge at every vertex.

Consequently, each Platonic solid can be characterized by an integer pair {p, q}. In this notation, p represents the number of edges (or, equivalently, vertices) on each face, while q denotes the number of faces (or, equivalently, edges) that converge at each vertex. This pair, {p, q}, is known as the Schläfli symbol and provides a combinatorial description of the polyhedron. The Schläfli symbols for the five Platonic solids are presented in the table below.

All other combinatorial data pertaining to these solids, including the total number of vertices (V), edges (E), and faces (F), can be derived from the values of p and q. Given that each edge connects two vertices and is shared by two adjacent faces, the following relationships must hold:

$${\displaystyle pF=2E=qV.\,}$$

Another fundamental relationship among these quantities is established by Euler's formula:

$${\displaystyle V-E+F=2.\,}$$

This assertion can be substantiated through various proofs. Collectively, these three relationships uniquely define the values of V, E, and F:

$${\displaystyle V={\frac {4p}{4-(p-2)(q-2)}}},\quad E={\frac {2pq}{4-(p-2)(q-2)}}},\quad F={\frac {4q}{4-(p-2)(q-2)}}.}$$

The interchange of p and q results in the reciprocal exchange of F and V, whereas E remains invariant.

Configuration Representation

Polyhedron elements can be represented within a configuration matrix, where rows and columns denote vertices, edges, and faces. Diagonal entries indicate the total count of each element within the polyhedron. Non-diagonal entries specify the number of column elements associated with or contained within the corresponding row element. Notably, the configuration matrices of dual polyhedra exhibit a 180-degree rotational symmetry with respect to each other.

Classification

A foundational result in geometry establishes the existence of only five convex regular polyhedra. Subsequent discussions present two prevalent arguments demonstrating that no more than five Platonic solids can exist; however, affirmatively proving the existence of a specific solid necessitates an explicit construction.

Geometric Proof

The subsequent geometric argument closely parallels that presented by Euclid in the Elements:

Topological Proof

A purely topological proof can be constructed utilizing solely combinatorial data pertaining to the solids. Central to this approach is Euler's observation that V − E + F = 2, alongside the relationship pF = 2E = qV, where p represents the number of edges per face and q denotes the number of edges incident to each vertex. The combination of these equations yields the following expression:

$${\displaystyle {\frac {2E}{q}}-E+{\frac {2E}{p}}=2.}$$

Subsequent simple algebraic manipulation yields the following:

§89§ q + §1819§ p = §2829§ §3031§ + §3839§ E . {\displaystyle {1 \over q}+{1 \over p}={1 \over 2}+{1 \over E}.}

Given that E is strictly positive, it necessarily follows that:

9§ q + §1819§ p > §2829§ §30

Considering that both p and q must be greater than or equal to 3, it is evident that only five distinct possibilities exist for the set {p, q}:

Geometric Properties

Angles

Each Platonic solid possesses several associated angles. The dihedral angle, defined as the interior angle formed between any two adjacent face planes, is denoted by θ for a solid {p,q} and can be calculated using the following formula:

$${\displaystyle \sin(\theta /2)={\frac {\cos(\pi /q)}{\sin(\pi /p)}}.}$$

Alternatively, this relationship can be more conveniently expressed using the tangent function as:

$${\displaystyle \tan(\theta /2)={\frac {\cos(\pi /q)}{\sin(\pi /h)}}.}$$

The value h, known as the Coxeter number, corresponds to 4 for the tetrahedron, 6 for both the cube and the octahedron, and 10 for both the dodecahedron and the icosahedron.

The angular deficiency at a polyhedron's vertex is defined as the discrepancy between the sum of its incident face-angles and 2π radians. For any vertex of the Platonic solids, denoted as {p,q}, this defect, δ, is calculated as:

δ<!-- δ --> = §1112§ π<!-- π --> −<!-- − --> q π<!-- π --> ( §2829§ − §3536§ p ) . {\displaystyle \delta =2\pi -q\pi \left(1-{2 \over p}\right).}

According to Descartes' theorem, this angular deficiency is equivalent to 4π divided by the total number of vertices, implying that the cumulative defect across all vertices sums to 4π.

The three-dimensional counterpart to a plane angle is the solid angle. For a Platonic solid, the solid angle, denoted as Ω, at each vertex can be expressed in relation to the dihedral angle as follows:

$${\displaystyle \Omega =q\theta -(q-2)\pi .\,}$$

This relationship is derived from the spherical excess formula applicable to spherical polygons, considering that the vertex figure of a polyhedron {p,q} constitutes a regular q-gon.

The solid angle subtended by a face from the center of a Platonic solid is equivalent to the solid angle of a complete sphere (4π steradians) divided by the total number of faces. Notably, this value also corresponds to the angular deficiency of the solid's dual polyhedron.

A comprehensive tabulation of the various angles pertinent to the Platonic solids is provided subsequently. All numerical values for solid angles are expressed in steradians. The constant φ, defined as ⁠1 + √§67§/§1213§⁠, represents the golden ratio.

Radii, Surface Area, and Volume

A significant characteristic of their regularity is that all Platonic solids invariably feature three concentric spheres:

• the circumscribed sphere, which encompasses the polyhedron and passes through all its vertices;
• the midsphere, which is tangent to every edge precisely at its midpoint; and
• the inscribed sphere, which is tangent to each face at its geometric center.

The respective radii of these spheres are designated as the circumradius, the midradius, and the inradius. These correspond to the distances from the polyhedron's center to its vertices, edge midpoints, and face centers, respectively. For a solid {p, q} with an edge length of a, the circumradius R and the inradius r are determined by the following expressions:

$${\displaystyle {\begin{aligned}R&={\frac {a}{2}}\tan \left({\frac {\pi }{q}}\right)\tan \left({\frac {\theta }{2}}\right)\\[3pt]r&={\frac {a}{2}}\cot \left({\frac {\pi }{p}}\right)\tan \left({\frac {\theta }{2}}\right)\end{aligned}}}$$

Here, θ represents the dihedral angle. The midradius, denoted as ρ, is calculated using the following formula:

$${\displaystyle \rho ={\frac {a}{2}}\cos \left({\frac {\pi }{p}}\right)\,{\csc }{\biggl (}{\frac {\pi }{h}}{\biggr )}}$$

In this context, h refers to the quantity previously defined in the dihedral angle's definition, with possible values being h = 4, 6, 6, 10, or 10. Notably, the ratio between the circumradius and the inradius exhibits symmetry with respect to p and q:

$${\displaystyle {\frac {R}{r}}=\tan \left({\frac {\pi }{p}}\right)\tan \left({\frac {\pi }{q}}\right)={\frac {\sqrt {{\csc ^{2}}{\Bigl (}{\frac {\theta }{2}}{\Bigr )}-{\cos ^{2}}{\Bigl (}{\frac {\alpha }{2}}{\Bigr )}}}{\sin {\Bigl (}{\frac {\alpha }{2}}{\Bigr )}}}.}$$

The surface area, A, of a Platonic solid {p, q} is derived from the product of the area of a regular p-gon and the total number of faces, F. This relationship is expressed by:

$${\displaystyle A={\biggl (}{\frac {a}{2}}{\biggr )}^{2}Fp\cot \left({\frac {\pi }{p}}\right).}$$

The volume is determined by the product of the number of faces, F, and the volume of a pyramid. This pyramid has a base defined by a regular p-gon and an inradius of r serving as its height. Specifically,

V = §1213§ §1415§ r A . {\displaystyle V={\frac {1}{3}}rA.}

A comprehensive listing is provided in the subsequent table, detailing the distinct radii, surface areas, and volumes associated with each Platonic solid. For standardization, the edge length, denoted as a, is set to a value of 2.

The constants φ and ξ referenced herein are defined as

φ<!-- φ --> = §1112§ cos ⁡<!-- ⁡ --> π<!-- π --> §2324§ = §3233§ + §3839§ §4344§ , ξ = §5657§ sin ⁡<!-- ⁡ --> π<!-- π --> §6869§ = §7879§ −<!-- − --> §8586§ §9091§ = §99100§ −<!-- − --> φ<!-- φ --> . {\displaystyle \varphi =2\cos {\pi \over 5}={\frac {1+{\sqrt {5}}}{2}},\qquad \xi =2\sin {\pi \over 5}={\sqrt {\frac {5-{\sqrt {5}}}{2}}}={\sqrt {3-\varphi }}.}

Within the category of Platonic solids, both the dodecahedron and the icosahedron are considered optimal approximations of a sphere. The icosahedron distinguishes itself by possessing the highest number of faces and the largest dihedral angle, resulting in the most precise embrace of its inscribed sphere. Furthermore, its surface area-to-volume ratio most closely aligns with that of a sphere of equivalent dimensions, whether measured by surface area or volume. Conversely, the dodecahedron is characterized by the minimal angular defect, the maximal vertex solid angle, and its superior capacity to encompass its circumscribed sphere.

Point in space

Considering an arbitrary point situated within the spatial domain of a Platonic solid, which possesses a circumradius denoted by R, its respective distances to the centroid of the solid and to each of its n vertices are represented by L and d_i. Under these conditions, and

S [ n ] ( §2021§ m ) = §3233§ n ∑ i = §4748§ n d i §6364§ m {\displaystyle S_{[n]}^{(2m)}={\frac {1}{n}}\sum _{i=1}^{n}d_{i}^{2m}} ,

the following relationship is established:

$${\displaystyle S_{[n]}^{(4)}+{\frac {16}{9}}R^{4}=\left(S_{[n]}^{(2)}+{\frac {2}{3}}R^{2}\right)^{2}.}$$

If d_i represents the distances from the n vertices of a Platonic solid to an arbitrary point on its circumscribed sphere, then the following relationship holds:

§67§ ( ∑<!-- ∑ --> i = §2223§ n d i §3839§ ) §4748§ = §5354§ n ∑ i = §6667§ n d i §8283§ . {\displaystyle 4\left(\sum _{i=1}^{n}d_{i}^{2}\right)^{2}=3n\sum _{i=1}^{n}d_{i}^{4}.}

Rupert Property

A polyhedron, denoted as P, possesses the Rupert property if another polyhedron, identical in shape to P but of equal or greater size, is capable of passing through an aperture within P. This characteristic is observed in all five Platonic solids.

Symmetry

Dual Polyhedra

Each polyhedron possesses a dual, also referred to as a 'polar' polyhedron, characterized by the interchange of its faces and vertices with faces and vertices interchanged. Notably, the dual of every Platonic solid is itself another Platonic solid, enabling the classification of the five solids into distinct dual pairs.

• The tetrahedron exhibits self-duality, meaning its dual form is also a tetrahedron.
• The cube and the octahedron constitute a dual pair.
• The dodecahedron and the icosahedron also constitute a dual pair.

If a polyhedron is represented by the Schläfli symbol {p, q}, its corresponding dual polyhedron will possess the symbol {q, p}. This relationship implies that each combinatorial property of a Platonic solid can be reinterpreted as an analogous combinatorial property of its dual.

The dual polyhedron can be constructed by designating the centers of the original figure's faces as the vertices of the dual. Subsequently, connecting the centers of adjacent faces in the original polyhedron establishes the edges of the dual, thereby interchanging the counts of faces and vertices while preserving the total number of edges.

More broadly, a Platonic solid can be dualized with respect to a sphere of radius d that is concentric with the solid. The radii of the original solid (R, ρ, r) and those of its dual (R*, ρ*, r*) are interconnected through the following relationship:

$${\displaystyle d^{2}=R^{\ast }r=r^{\ast }R=\rho ^{\ast }\rho .}$$

Dualization with respect to the midsphere (d = ρ) is frequently advantageous, as the midsphere maintains an identical relationship with both polyhedra. When d§67§ = Rr is applied, the resulting dual solid possesses equivalent circumradius and inradius, specifically R* = R and r* = r.

Symmetry Groups

Within mathematics, the concept of symmetry is rigorously examined through the framework of a mathematical group. Each polyhedron is characterized by an associated symmetry group, comprising the complete set of transformations (Euclidean isometries) that preserve the polyhedron's invariant form. The cardinality of this symmetry group corresponds to the total number of symmetries exhibited by the polyhedron. A distinction is commonly drawn between the full symmetry group, which encompasses reflections, and the proper symmetry group, which is restricted solely to rotations.

The symmetry groups pertinent to the Platonic solids constitute a distinct category of three-dimensional point groups, specifically termed polyhedral groups. The pronounced symmetry inherent in Platonic solids can be understood through various perspectives. Crucially, all vertices, edges, and faces of each solid are equivalent under the operation of its symmetry group. This implies that the symmetry group's action is transitive across the vertices, edges, and faces. Consequently, this property offers an alternative definition for polyhedral regularity: a polyhedron is deemed regular if and only if it exhibits uniformity across its vertices, edges, and faces.

Only three distinct symmetry groups are associated with the Platonic solids, not five, because the symmetry group of any given polyhedron is identical to that of its dual. This equivalence becomes evident upon analyzing the construction process of a dual polyhedron, where any symmetry present in the original polyhedron must also be a symmetry of its dual, and conversely. The three identified polyhedral groups are:

• the tetrahedral group, denoted as T,
• the octahedral group, denoted as O (which additionally serves as the symmetry group for the cube), and
• the icosahedral group, denoted as I (which also represents the symmetry group of the dodecahedron).

The orders of the proper (rotational) groups are 12, 24, and 60, respectively, which precisely corresponds to twice the number of edges in their corresponding polyhedra. The orders of the full symmetry groups are subsequently doubled, yielding values of 24, 48, and 120. A detailed derivation of these properties can be found in (Coxeter 1973). Furthermore, all Platonic solids, with the exception of the tetrahedron, exhibit central symmetry, indicating their invariance under reflection through the origin.

The subsequent table enumerates the diverse symmetry properties characteristic of the Platonic solids. The symmetry groups presented represent the full groups, with their rotational subgroups indicated parenthetically (a similar notation applies to the number of symmetries). Wythoff's kaleidoscope construction provides a methodology for directly generating polyhedra from their respective symmetry groups. For reference, Wythoff's symbol is provided for each Platonic solid.

In Nature and Technology

The tetrahedron, cube, and octahedron are all observed to occur naturally within various crystal structures. However, these forms do not encompass the entirety of possible crystal morphologies. Notably, neither the regular icosahedron nor the regular dodecahedron are found among them. One specific form, designated as the pyritohedron (named after the mineral group it typifies), features twelve pentagonal faces arranged in a configuration identical to that of the regular dodecahedron. Nevertheless, the faces of the pyritohedron are not regular, thus rendering the pyritohedron itself irregular. Allotropes of boron and numerous boron compounds, including boron carbide, incorporate discrete B₁₂ icosahedra within their crystalline architectures. Similarly, carborane acids exhibit molecular structures that approximate regular icosahedra.

In the early 20th century, Ernst Haeckel documented several Radiolaria species exhibiting skeletons configured as various regular polyhedra. Notable examples encompass Circoporus octahedrus, Circogonia icosahedra, Lithocubus geometricus, and Circorrhegma dodecahedra, with their morphological characteristics evident from their nomenclature.

Numerous viruses, including the herpes virus, adopt the morphology of a regular icosahedron. Viral capsids are constructed from repetitive, identical protein subunits, and the icosahedron represents the most efficient configuration for assembly using these components. The utilization of a regular polyhedron facilitates construction from a single, repeatedly employed basic protein unit, thereby conserving space within the viral genome.

Within meteorology and climatology, there is growing interest in global numerical models of atmospheric flow that incorporate geodesic grids derived from an icosahedron (subsequently refined through triangulation), as opposed to the conventional longitude/latitude grid. This approach offers the benefit of uniformly distributed spatial resolution, devoid of singularities such as the poles, albeit at the cost of increased numerical complexity.

The geometric design of space frames frequently incorporates Platonic solids. Within the MERO system, Platonic solids serve as a naming convention for diverse space frame configurations. For instance, ⁠§23§/§67§⁠O+T denotes a configuration comprising half an octahedron and a tetrahedron.

Various Platonic hydrocarbons, such as cubane and dodecahedrane, have been successfully synthesized.

Liquid Crystals Exhibiting Platonic Solid Symmetries

The presence of such symmetries in the intermediate material phase known as liquid crystals was initially theorized in 1981 by H. Kleinert and K. Maki. Subsequently, Dan Shechtman discovered the icosahedral structure in aluminum three years later, an achievement for which he was awarded the Nobel Prize in Chemistry in 2011.

Cultural Applications

Platonic solids are frequently employed in the manufacturing of dice due to their capacity to produce fair outcomes. While six-sided dice are ubiquitous, other polyhedral forms are routinely utilized in role-playing games. These dice are conventionally designated as dn, where n signifies the number of faces (e.g., d8, d20).

These geometric forms commonly appear in various other games and puzzles. Puzzles analogous to a Rubik's Cube are available in all five Platonic solid configurations.

Architectural Applications

Architects were drawn to Plato's concept of timeless forms, perceivable by the soul within material objects; however, they adapted these shapes into more construction-appropriate geometries such as the sphere, cylinder, cone, and square pyramid. Notably, Étienne-Louis Boullée, a prominent figure in neoclassicism, extensively explored an architectural interpretation of "Platonic solids."

Related Polyhedra and Polytopes

Uniform Polyhedra

Four non-convex regular polyhedra, known as Kepler–Poinsot polyhedra, exist. All possess icosahedral symmetry and can be derived as stellations of the dodecahedron and the icosahedron.

Following the Platonic solids, the subsequent most regular convex polyhedra include the cuboctahedron, which represents a rectification of the cube and the octahedron, and the icosidodecahedron, a rectification of the dodecahedron and the icosahedron (the rectification of the self-dual tetrahedron yields a regular octahedron). Both are classified as quasi-regular, indicating their vertex- and edge-uniformity and regular faces, despite the faces not being entirely congruent (occurring in two distinct classes). These two polyhedra constitute a portion of the thirteen Archimedean solids, which are convex uniform polyhedra exhibiting polyhedral symmetry. Their duals, the rhombic dodecahedron and rhombic triacontahedron, are edge- and face-transitive; however, their faces are not regular, and their vertices each present in two distinct types, classifying them as two of the thirteen Catalan solids.

Uniform polyhedra constitute a significantly broader category of polyhedral forms. These figures are characterized by vertex-uniformity and possess one or more types of regular or star polygonal faces. This class encompasses all previously mentioned polyhedra, alongside an infinite series of prisms, an infinite series of antiprisms, and 53 additional non-convex configurations.

Johnson solids are defined as convex polyhedra featuring regular faces but lacking uniformity. This category includes five of the eight convex deltahedra, which possess identical, regular faces composed entirely of equilateral triangles, yet are not uniform. The remaining three convex deltahedra are the Platonic tetrahedron, octahedron, and icosahedron.

Regular Tessellations

The three regular tessellations of a plane exhibit a close relationship with the Platonic solids. Specifically, Platonic solids can be conceptualized as regular tessellations of a sphere. This conceptualization involves projecting each solid onto a concentric sphere, where its faces transform into regular spherical polygons that precisely cover the sphere's surface. Spherical tilings introduce two additional infinite families of regular tilings: hosohedra, denoted as {2,n}, characterized by two vertices at the poles and lune-shaped faces; and their duals, dihedra, denoted as {n,2}, which feature two hemispherical faces and vertices regularly distributed along the equator. These particular tessellations would be considered degenerate if represented as polyhedra in three-dimensional space.

Each regular tessellation of a sphere is defined by an ordered pair of integers {p, q}, satisfying the condition ⁠§67§/p⁠ + ⁠§1819§/q⁠ > ⁠§3031§/§3435§⁠. Similarly, a regular tessellation of a plane is characterized by the condition ⁠§4041§/p⁠ + ⁠§5253§/q⁠ = ⁠§6465§/§6869§⁠. This latter condition presents three distinct possibilities.

Analogously, regular tessellations of the hyperbolic plane can be examined. These are defined by the condition ⁠§23§/p⁠ + ⁠§1415§/q⁠ < ⁠§2627§/§3031§⁠. This category encompasses an infinite family of such tessellations.

Higher Dimensions

Beyond three dimensions, polyhedra are generalized into polytopes, where higher-dimensional convex regular polytopes serve as the analogues of three-dimensional Platonic solids.

During the mid-19th century, Swiss mathematician Ludwig Schläfli identified the four-dimensional counterparts to the Platonic solids, termed convex regular 4-polytopes. Precisely six such figures exist: five are analogous to the Platonic solids—specifically, the 5-cell ({3,3,3}), the 16-cell ({3,3,4}), the 600-cell ({3,3,5}), the tesseract ({4,3,3}), and the 120-cell ({5,3,3})—while the sixth is the self-dual 24-cell ({3,4,3}).

For all dimensions exceeding four, only three convex regular polytopes are found: the simplex, represented as {3,3,...,3}; the hypercube, represented as {4,3,...,3}; and the cross-polytope, represented as {3,3,...,4}. In three dimensions, these correspond to the tetrahedron ({3,3}), the cube ({4,3}), and the octahedron ({3,4}), respectively.

Citations

Citations

General and Cited Sources

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  • Platonic solids at Encyclopaedia of Mathematics
  • Weisstein, Eric W. provides an entry on "Platonic solid" in MathWorld.
  • Weisstein, Eric W. provides an entry on "Isohedron" in MathWorld.
  • A discussion of Book XIII from Euclid's Elements.
  • Interactive three-dimensional polyhedra rendered in Java.
  • Platonic solids presented within Visual Polyhedra.
  • The Solid Body Viewer offers an interactive three-dimensional polyhedron viewing experience, enabling model export in SVG, STL, or OBJ formats.
  • Interactive folding and unfolding representations of Platonic solids, archived and implemented in Java.
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  • Freely available paper models (nets) of Platonic solids.
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