# Crystallography

This is a single crystal XRD unit. Credit: Anjali Merin.

Crystallography is the science that examines crystals, which can be found everywhere in nature—from salt to snowflakes to gemstones.

It uses the properties and inner structures of crystals to determine the arrangement of atoms and generate knowledge.

## Crystallography and Mineral Evolution

Within the past century, crystallography has been a primary force in driving major advances in the detailed understanding of materials, synthetic chemistry, the understanding of basic principles of biological processes, genetics, and has contributed to major advances in the development of drugs for numerous diseases. It is one of the most important tool to study solids, since most of the materials in solid state exhibits crystalline nature. Crystalline solids are known to show different structural forms depending on different conditions of temperature, pressure etc. Since minerals are naturally occurring inorganic crystalline, the phase transitions involving minerals will be interesting. Phase transition studies of minerals deserve special attention as they can provide clues to mechanism of mineral evolution on earth crust. They also opens up the possibility of utilizing the naturally abundant minerals for generating novel functional materials processing properties such as ionic conductivity, ferroelecticity, ferromagnetism etc .[1][1]

A mineral is an element or chemical compound that is normally crystalline and that has been formed as a result of geological process. It has highly ordered atomic structure and specific physical properties . Mimicking the mineral evolution process in laboratory by phase transition studies can throw lights on our understanding of mineral evolution. A large number of minerals occur in hydrated form, especially the bimetallic sulfate minerals are widely interested in its varying non stoichiometric crystal structure with the levels of hydration. Bimetallic sulfates are more interested due to their phase transition with temperature, eg; langbenites, krohnkite. The non-stoichiometric structures getting trapped in small kinetically stabilized energy wells, which are intermediates between individual monosulfates and bimetallic sulfate minerals, may have valuable structural hints about the process of origin of such minerals. The phase transition studies reveal that those structures are precursors of the original mineral and the ubiquitous role of water in the formation of minerals in the earth crust .[2][2]

In particular, Langebenite minerals are distinctive geological minerals found in only a few locations in the world. These deposits were formed millions of years ago when a variety of salts were left behind after the evaporation of ancient ocean beds. These type crystals have general chemical formulae A2B2(SO4)3 where A denotes a monovalent cation such as K, NH4… and B a divalent cation such as Mg, Mn, Ni… At high temperatures, they crystallizes isomorphously in the cubic space group P213. They are having a wide variety of applications due to its ferroelectric, ferroelastic, spectroscopic and magnetic properties [2]. Their best-known applications are in dosimetry of ionizing radiation, CTV screen phosphors, projection T V phosphors, scintillators, fluorescent lamps, full color displays, X-ray storage and screens intensifying phosphors, and laser materials .[3][3]

## Theoretical crystallography

Def. an "experimental science of determining the arrangement of atoms in solids"[4], or the "study of crystals"[4] is called crystallography.

## Formula units

This is a model of how NaCl formula units could form a cube. Credit: BruceBlaus.

Illustration shows the close-packing of equal spheres in both hcp (left) and fcc (right) lattices. Credit: Twisp.

Flame emission spectroscopy of a test mineral described in the above section suggests some conclusions about the mineral.

The halide present was chlorine. The mineral is most likely halite. The formula unit is NaCl. The mineral grains appear to be cuboidal. At the left is a model of how NaCl formula units could form a cube.

When NaCl is dissolved in water, it has the formula [Na(H2O)8]+. The chloride ion is surrounded by an average of 6 molecules of water.[5] As the water evaporates, the cations of sodium and the anions of chlorine should be drawn back together.

If the sodium and chlorine ions can be represented by equal-sized hard balls, they would be expected to form close-packed solids.

An examination of the two types of close-packed structures shows a problem that may disqualify such structures for representing NaCl. Each sphere is the same distance from every other sphere. An effort to use some as sodiums and the others as chlorines always results in at least two sodiums contacting each other and the same thing happens with the chlorines.

## Models

This is a body-centered cubic unit cell. Credit: Baszoetekouw.

Crystal structure of NaCl shows coordination polyhedra. Credit: Solid State.

Relative radii of atoms and ions for neutral atoms colored gray, cations red, and anions blue. Credit: Popnose.
Common dodecahedra
Ih, order 120
Regular- Small stellated- Great- Great stellated-

Th, order 24 T, order 12 Oh, order 48 Td, order 24
Pyritohedron Tetartoid Rhombic- Trapezoidal-

D4h, order 16 D3h, order 12
Rhombo-hexagonal- Rhombo-square- Trapezo-rhombic- Rhombo-triangular-

A slightly more open structure is the body-centered cubic shown in the image on the left. Here, one sodium ion could be surrounded by six chlorines in an octahedron, and one chlorine anion could be surrounded by six sodium cations. These are shown in the second image down on the left.

Using the approximate ionic radii from the third image down on the right for Na+ as 116 pm and 167 pm for Cl- to calculate a radius ratio yields 0.695. Such a size ratio falls in the octahedron range of ≥ 0.414 and < 0.732.

The ball and stick model on the left shows what's inside the cube.

## Habits

The image shows that halite can occur in a massive habit. Credit: Helix84.

Def. form "of growth or general appearance of a variety or species of plant or crystal"[6] is called a habit.

The image on the left shows that halite can occur in a massive habit but is apparently always crystalline.

Habit[7][8][9] Image Description Common Example(s)
Acicular   Needle-like, slender and/or tapered Natrolite, Rutile[10]
Amygdule (Amygdaloidal)   Almond-shaped Heulandite, subhedral Zircon
Botryoidal or globular   Grape-like, hemispherical masses Hematite, Pyrite, Malachite, Smithsonite, Hemimorphite, Adamit, Variscite
Columnar   Similar to fibrous: Long, slender prisms often with parallel growth Calcite, Gypsum/Selenite
Coxcomb   Aggregated flaky or tabular crystals closely spaced. Barite, Marcasite
Cubic   Cube shape Pyrite, Galena, Halite
Dendritic or arborescent   Tree-like, branching in one or more direction from central point Romanechite and other Manganese (Mn)-oxide minerals, magnesite, native copper
Dodecahedral   Rhombic dodecahedron, 12-sided Garnet
Drusy or encrustation   Aggregate of minute crystals coating a surface or cavity Uvarovite, Malachite, Azurite, Quartz
Enantiomorphic   Mirror-image habit (i.e. crystal twinning) and optical characteristics; right- and left-handed crystals Quartz, Plagioclase, Staurolite
Equant, stout   Length, width, and breadth roughly equal Olivine, Garnet
Fibrous   Extremely slender prisms Serpentine group, Tremolite (i.e. Asbestos)
Filiform or capillary   Hair-like or thread-like, extremely fine many Zeolites
Foliated or micaceous or lamellar (layered)   Layered structure, parting into thin sheets Mica (Muscovite, Biotite, etc.)
Granular   Aggregates of anhedral crystals in matrix Bornite, Scheelite
Hemimorphic   Doubly terminated crystal with two differently shaped ends Hemimorphite, Elbaite
Hexagonal   Hexagon shape, six-sided Quartz, Hanksite
Hopper crystals   Like cubic, but outer portions of cubes grow faster than inner portions, creating a concavity Halite, Calcite, synthetic Bismuth
Mammillary   Breast-like: surface formed by intersecting partial spherical shapes, larger version of botryoidal, also concentric layered aggregates Malachite, Hematite
Massive or compact   Shapeless, no distinctive external crystal shape Limonite, Turquoise, Cinnabar, Realgar
Nodular or tuberose   Deposit of roughly spherical form with irregular protuberances Chalcedony, various Geodes
Octahedral   Octahedron, eight-sided (two pyramids base to base) Diamond, Magnetite
Platy   Flat, tablet-shaped, prominent pinnacoid Wulfenite
Plumose   Fine, feather-like scales Aurichalcite, Boulangerite, Mottramite
Prismatic   Elongate, prism-like: well-developed crystal faces parallel to the vertical axis Tourmaline, Beryl
Pseudo-hexagonal   Hexagonal appearance due to cyclic twinning Aragonite, Chrysoberyl
Radiating or divergent   Radiating outward from a central point Wavellite, Pyrite suns
Reniform or colloform   Similar to botryoidal/mamillary: intersecting kidney-shaped masses Hematite, Pyrolusite, Greenockite
Reticulated   Crystals forming net-like intergrowths Cerussite
Rosette or lenticular (lens shaped crystals)   Platy, radiating rose-like aggregate Gypsum, Barite (i.e. Desert rose)
Sphenoid   Wedge-shaped Sphene
Stalactitic   Forming as stalactites or stalagmites; cylindrical or cone-shaped Calcite, Goethite, Malachite
Striated   Not a habit per se, but a condition of lines that can grow on certain crystal faces on certain minerals Tourmaline, Pyrite, Quartz, Feldspar, Sphalerite
Stubby or blocky or tabular   More elongated than equant, slightly longer than wide, flat tablet shaped Feldspar, Topaz
Tetrahedral   Tetrahedra-shaped crystals Tetrahedrite, Spinel, Magnetite
Wheat sheaf   Aggregates resembling hand-reaped wheat sheaves Stilbite

## Cleavages

Image shows a mineral specimen of Halite. Credit: Vassia Atanassova - Spiritia.

The image on the right shows a more familiar crystal habit of halite.

Halite might be expected to break leaving behind a cube-like face.

Def. the "tendency of a crystal to split along specific planes"[11] is called cleavage.

The cleavage described for halite is "{001} perfect Fracture conchoidal. Brittle."[12]

Miller indices {h k ℓ} describe potential cleavage planes. Credit: DeepKling.

Cleavage forms parallel to crystallographic planes:[13]

• Basal or pinacoidal cleavage occurs when there is only one cleavage plane. Graphite has basal cleavage. Mica (like muscovite or biotite) also has basal cleavage; this is why mica can be peeled into thin sheets.

Meroxene biotite
• Cubic cleavage occurs on when there are three cleavage planes intersecting at 90 degrees. Halite (or salt) has cubic cleavage, and therefore, when halite crystals are broken, they will form more cubes.

Halite
• Octahedral cleavage occurs when there are four cleavage planes in a crystal. Fluorite exhibits perfect octahedral cleavage. Octahedral cleavage is common for semiconductors. Diamond also has octahedral cleavage.

Fluorite
• Rhombohedral cleavage occurs when there are three cleavage planes intersecting at angles that are not 90 degrees. Calcite has rhombohedral cleavage.

Calcite
• Prismatic cleavage occurs when there are two cleavage planes in a crystal. Spodumene exhibits prismatic cleavage.

Spodumene
• Dodecahedral cleavage occurs when there are six cleavage planes in a crystal. Sphalerite has dodecahedral cleavage.

Pyrite

## Unit cells

This is a possible unit cell for crystalline NaCl. Credit: Ed Caruthers.

Def. the "smallest repeating structure (parallelepiped) of atoms within a crystal, from which the structure of the complete crystal can be inferred"[14] is called a unit cell.

An examination of the model unit cell for crystalline NaCl shows a repeat pattern along the left front edge starting with a Cl atom at the corner. Moving along the line of atoms at this lowest edge, next is a smaller Na atom, then another Cl atom. This second Cl atom is a repeat of the first.

Going back to the corner Cl atom and moving straight up above it is a Na atom. Above that Na atom is again another Cl that is another repeat of the first.

Looking at the eight corners of this perspective view of the model there is a Cl atom at each corner. Starting again at the lower left corner Cl, directly behind the left rear face of this unit cell is another such unit cell not shown. Further left of these two unit cells are two more. One is along the base diagonal though the corner Cl and the second is in front sharing the face of the model unit cell shown. Counting the model shown there are four unit cells in this layer that share the lower left corner Cl atom.

Right below this layer of four unit cells are four identical unit cells all cornered to this same Cl atom in the lower left corner of the model. Summarizing, the corner of a unit cell is shared by eight unit cells total.

From visualizing nearer unit cells next to the one drawn, an edge of this cell is shared by four unit cells, and each face by two unit cells.

Looking at the numbers of Cls and Nas:

1. eight corner Cls are each shared by eight unit cells so one corner Cl per unit cell,
2. along each edge between each corner Cl is a Na, each of 12 edges is shared by four unit cells, so 3 Nas per unit cell,
3. each face shares a centered Cl with two unit cells, 6 such Cls, each shared by two unit cells, is 3 Cls,
4. inside the unit cell pictured there is one and only one Na, shared with no other unit cell, so one Na,

totalling the anions, there are 4 Cls, totalling the cations there are 4 Nas. In a unit cell for this model of halite, there are four formula units. The formula content of a unit cell is often denoted as Z. For this model, Z = 4.

## Structures

Even when the mineral grains are too small to see or are irregularly shaped, the underlying crystal structure is always periodic and can be determined by X-ray diffraction.[15] Minerals are typically described by their symmetry content. Crystals are restricted to 32 point groups, which differ by their symmetry. These groups are classified in turn into more broad categories, the most encompassing of these being the six crystal families.[16] All of these six crystal families when combined with the 32 point groups result in 230 space groups that symmetrically describe all space-filling three-dimensional crystal structures.

These families can be described by the relative lengths of the three crystallographic axes, and the angles between them; these relationships correspond to the symmetry operations that define the narrower point groups. They are summarized below; a, b, and c represent the axes, and α, β, γ represent the angle opposite the respective crystallographic axis (e.g. α is the angle opposite the a-axis, viz. the angle between the b and c axes):[16]

Crystal family Lengths Angles Common mineral examples
Isometric (Cubic crystal system) a=b=c α=β=γ=90° garnet, halite, pyrite
Tetragonal a=b≠c α=β=γ=90° rutile, zircon, andalusite
Orthorhombic a≠b≠c α=β=γ=90° olivine, aragonite, orthopyroxenes
Hexagonal a=b≠c α=β=90°, γ=120° Quartz, calcite, tourmaline
Monoclinic a≠b≠c α=γ=90°, β≠90° clinopyroxenes, orthoclase, gypsum
Triclinic a≠b≠c α≠β≠γ≠90° Anorthite, albite, kyanite
Crystal Family Lattice System Schönflies notation 14 Bravais Lattices
Primitive (P) Base-centered (C) Body-centered (I) Face-centered (F)
Triclinic Ci
Monoclinic C2h
Orthorhombic D2h
Tetragonal D4h
Hexagonal Rhombohedral D3d
Hexagonal D6h
Cubic Oh

The unit cells are specified according to the relative lengths of the cell edges (a, b, c) and the angles between them (α, β, γ). The volume of the unit cell can be calculated by evaluating the triple product a · (b × c), where a, b, and c are the lattice vectors. The properties of the lattice systems are given below:

Crystal family Lattice system Volume Axial distances (edge lengths)[17] Axial angles[17] Corresponding examples
Triclinic ${\displaystyle abc{\sqrt {1-\cos ^{2}\alpha -\cos ^{2}\beta -\cos ^{2}\gamma +2\cos \alpha \cos \beta \cos \gamma }}}$  (All remaining cases) potassium dichromate (K2Cr2O7), copper(II) sulfate (CuSO4·5H2O), boric acid (H3BO3)
Monoclinic ${\displaystyle abc\,\sin \beta }$  ac α = γ = 90°, β ≠ 90° monoclinic sulphur, sodium sulfate (Na2SO4·10H2O), lead(II) chromate (PbCrO3)
Orthorhombic ${\displaystyle abc}$  abc α = β = γ = 90° rhombic sulphur, potassium nitrate (KNO3), barium sulfate (BaSO4)
Tetragonal ${\displaystyle a^{2}c}$  a = bc α = β = γ = 90° white tin, tin dioxide (SnO2), titanium dioxide (TiO2), calcium sulfate (CaSO4)
Hexagonal Rhombohedral ${\displaystyle a^{3}{\sqrt {1-3\cos ^{2}\alpha +2\cos ^{3}\alpha }}}$  a = b = c α = β = γ ≠ 90° calcite (CaCO3), cinnabar (HgS)
Hexagonal ${\displaystyle {\frac {\sqrt {3}}{2}}\,a^{2}c}$  a = b α = β = 90°, γ = 120° graphite, zinc oxide (ZnO), cadmium sulphide (CdS)
Cubic ${\displaystyle a^{3}}$  a = b = c α = β = γ = 90° sodium chloride (NaCl), zinc blende, copper metal, potassium chloride (KCl), diamond, silver metal

## Visual crystallography

The image shows an example of Mitscherlich's goniometer. Credit: Samuel Orgelbrand's Universal Encyclopedia.

## Geometry of the 24-cell

The 24-cell is the convex regular 4-polytope[18] with Schläfli symbol {3,4,3}.

The boundary of the 24-cell is composed of 24 octahedral cells with six meeting at each vertex, and three at each edge. Together they have 96 triangular faces, 96 edges, and 24 vertices. The vertex figure is a cube. The 24-cell is self-dual. In fact, the 24-cell is the unique convex self-dual regular Euclidean polytope which is neither a polygon nor a simplex. It is the only one of the six convex regular 4-polytopes which is not the dimensional analogue of one of the five Platonic solids. It has no regular analogue in 3 dimensions, but it can be considered the analogue of a dual pair of irregular solids: the cuboctahedron and its dual the rhombic dodecahedron.

The 24-cell is the symmetric union of the geometries of every convex regular polytope in the first four dimensions, except those with a 5 in their Schlӓfli symbol.[a] It is especially useful to explore the 24-cell, because one can see all the geometric relationships among all of those polytopes in a single 24-cell or its honeycomb.

The 24-cell is the fourth in the sequence of 6 convex regular 4-polytopes (in order of size and complexity).[b] It can be deconstructed into 3 overlapping instances of its predecessor the tesseract (8-cell), as the 8-cell can be deconstructed into 2 overlapping instances of its predecessor the 16-cell.[20] The reverse procedure to construct each of these from an instance of its predecessor preserves the radius of the predecessor, but generally produces a successor with a different edge length.[d]

### Coordinates

The 24-cell is the convex hull of its vertices. The Cartesian coordinates[21] of its vertices may be given in several different forms, depending on our choice of coordinate system. Each form of coordinates best illustrates a different aspect of the vertex geometry.

#### Squares

The 24-cell can be described as the 24 coordinate permutations of:

${\displaystyle (\pm 1,\pm 1,0,0)\in \mathbb {R} ^{4}}$ .

In this form the 24-cell has edges of length 2 and is inscribed in a 3-sphere of radius 2. Remarkably, the edge length equals the circumradius, as in the hexagon, or the cuboctahedron. Such polytopes are radially equilateral.[c]

The 24 vertices can be seen as the vertices of 6 orthogonal[e] squares[f] which intersect[g] only at the their common center.

#### Hexagons

The 24-cell is self-dual, having the same number of vertices (24) as cells and the same number of edges (96) as faces.

If the dual of the above 24-cell of edge length 2 is taken by reciprocating it about its inscribed sphere, another 24-cell is found which has edge length and circumradius 1, and its coordinates reveal more structure. In this form the vertices of the 24-cell can be given as follows:

8 vertices obtained by permuting the integer coordinates:

(±1, 0, 0, 0)

and 16 vertices with half-integer coordinates of the form:

1/2, ±1/2, ±1/2, ±1/2)

all 24 of which lie at distance 1 from the origin.[c]

Viewed as quaternions, these are the unit Hurwitz quaternions.

The 8 integer coordinates can be seen as the vertices of 6 perpendicular[e] squares[h] which intersect[g] only at the their common center.

The 24 vertices are the vertices of 4 orthogonal hexagons[i] which intersect only at their common center.[j]

#### Triangles

The 24 vertices are the vertices of 8 triangles in 4 orthogonal planes[k] which intersect only at their common center.

The 24 vertices are also the vertices of 96 other triangles[l] in 48 parallel pairs, in planes one unit length apart which do not pass through the center.[m]

If the 24-cell of edge length and circumradius 3 is taken,[n] its coordinates reveal more structure. In this form the vertices of the 24-cell can be given as follows:

${\displaystyle (0,\pm 1,\pm 1,\pm 1)\in \mathbb {R} ^{4}}$

and can be seen to be the vertices of 24 tetrahedra[o] inscribed in the 24-cell.[p]

#### Hypercubic chords

Vertex geometry of the radially equilateral[c] 24-cell, showing the 3 great circle polygons and the 4 vertex-to-vertex chord lengths.

The 24 vertices of the 24-cell are distributed[22] at four different chord lengths from each other: 1, 2, 3 and 4.

Each vertex is joined to 8 others[q] by an edge of length 1, spanning 60° = 𝜋/3 of arc. Next nearest are 6 vertices[r] located 90° = 𝜋/2 away, along an interior chord of length 2. Another 8 vertices lie 120° = 2𝜋/3 away, along an interior chord of length 3. The opposite vertex is 180° = 𝜋 away along a diameter of length 2. Finally, as the 24-cell is radially equilateral, its center can be treated[s] as a 25th canonical apex vertex[t], which is 1 edge length away from all the others.

To visualize how the interior polytopes of the 24-cell fit together (as described below), keep in mind that the four chord lengths (1, 2, 3, 4) are the long diameters of the hypercubes of dimensions 1 through 4: the long diameter of the square is 2; the long diameter of the cube is 3; and the long diameter of the tesseract is 4. Moreover, the long diameter of the octahedron is 2 like the square; and the long diameter of the 24-cell itself is 4 like the tesseract.

#### Geodesics

The vertex chords of the 24-cell are arranged in geodesic great circles which lie in sets of orthogonal planes. The geodesic distance between two 24-cell vertices along a path of 1 edges is always 1, 2, or 3, and it is 3 only for opposite vertices.[u]

The 1 edges occur in 16 hexagonal great circles (4 sets of 4 orthogonal[j] planes), 4 of which cross at each vertex[w]. The 96 distinct 1 edges divide the surface into 96 triangular faces and 24 octahedral cells: a 24-cell.

The 2 chords occur in 18 square great circles (3 sets of 6 orthogonal[e] planes), 3 of which cross at each vertex[x]. The 72 distinct 2 chords do not run in the same planes as the hexagonal great circles; they do not follow the 24-cell's edges, they pass through its cell centers below one of its mid-edges.[y]

The 3 chords occur in 32 triangular great circles in 16 planes (4 sets of 4 orthogonal planes), 4 of which cross at each vertex[z]. The 96 distinct 3 chords run vertex-to-every-other-vertex in the same planes as the hexagonal great circles[k].

The 4 chords occur as 12 vertex-to-vertex diameters (3 sets of 4 orthogonal axes), the 24 radii around the 25th central vertex[t].

The 1 edges occur in 48 parallel pairs, 3 apart. The 2 chords occur in 36 parallel pairs, 2 apart. The 3 chords occur in 48 parallel pairs, 1 apart.

Each great circle plane intersects[g] with each of the other great circle planes or face planes to which it is orthogonal at the center point only, and with each of the others to which it is not orthogonal at a single edge of some kind. In every case that edge is one of the vertex chords of the 24-cell.[ab]

### Constructions

Triangles and squares come together uniquely in the 24-cell to generate, as interior features[s], all of the triangle-faced and square-faced regular convex polytopes in the first four dimensions (with caveats for the 5-cell and the 600-cell).[ac] Consequently, there are numerous ways to construct or deconstruct the 24-cell.

#### Reciprocal constructions from 8-cell and 16-cell

The 8 integer vertices (±1, 0, 0, 0) are the vertices of a regular 16-cell, and the 16 half-integer vertices (±1/2, ±1/2, ±1/2, ±1/2) are the vertices of its dual, the tesseract (8-cell). The tesseract gives Gosset's construction[25] of the 24-cell, equivalent to cutting a tesseract into 8 cubic pyramids, and then attaching them to the facets of a second tesseract. The analogous construction in 3-space gives the rhombic dodecahedron which, however, is not regular. The 16-cell gives the reciprocal construction of the 24-cell, Cesaro's construction,[26] equivalent to rectifying a 16-cell (truncating its corners at the mid-edges, as described above). The analogous construction in 3-space gives the cuboctahedron (dual of the rhombic dodecahedron) which, however, is not regular. The tesseract and the 16-cell are the only regular 4-polytopes in the 24-cell.[27]

The vertices of the 24-cell can be grouped into three disjoint sets of eight with each set defining a regular 16-cell, and with the complement defining the dual tesseract.[28]

#### Truncations

We can truncate[29] the 24-cell by slicing through planes bounded by vertex chords to remove vertices, exposing the facets of interior 4-polytopes inscribed in the 24-cell. One can cut a 24-cell into two parts through any planar hexagon of 6 vertices, any planar square of 4 vertices, or any planar triangle of 3 vertices. The great circle planes (above) are only some of those planes. Here we shall expose some of the others: the face planes of interior polytopes, which divide the 24-cell into two unequal parts.[ad]

##### 8-cell

Starting with a complete 24-cell, remove 8 orthogonal vertices (4 opposite pairs on 4 perpendicular axes), and the 8 edges which radiate from each, by slicing through 24 square face planes bounded by 1 edges to remove 8 cubic pyramids whose apexes are the vertices to be removed. This removes 4 edges from each hexagonal great circle (retaining just one opposite pair of edges), so no continuous hexagonal great circles remain. Now 3 perpendicular edges meet and form the corner of a cube at each of the 16 remaining vertices[ae], and the 32 remaining edges divide the surface into 24 square faces and 8 cubic cells: a tesseract. There are three ways you can do this (choose a set of 8 orthogonal vertices out of 24), so there are three such tesseracts inscribed in the 24-cell. They overlap with each other, but most of their element sets are disjoint: they share some vertex count, but no edge length, face area, or cell volume; they do share 4-content.

##### 16-cell

Starting with a complete 24-cell, remove the 16 vertices of a tesseract (retaining the 8 vertices you removed above), by slicing through 32 triangular face planes bounded by 2 chords to remove 16 tetrahedral pyramids whose apexes are the vertices to be removed. This removes 12 square great circles (retaining just one orthogonal set) and all the 1 edges, exposing 2 chords as the new edges. Now the remaining 6 square great circles cross perpendicularly, 3 at each of 8 remaining vertices[af], and their 24 edges divide the surface into 32 triangular faces and 16 tetrahedral cells: a 16-cell. There are three ways you can do this (remove 1 of 3 sets of tesseract vertices), so there are three such 16-cells inscribed in the 24-cell. They do not overlap with each other, and all of their element sets are disjoint: they do not share any vertex count, edge length, face area, cell volume, or 4-content.

##### 5-cell

Starting with a complete 24-cell, remove 20 vertices, by slicing through 4 triangular face planes bounded by 3 chords to remove 5 vertices above each plane.[ag] This removes 20 triangular great circles, and all the 1 and 2 chords, exposing 3 chords as the new edges. Now the remaining 4 triangular great circles meet but do not cross, 3 at each of the 4 remaining vertices[ah], and their 6 edges divide the surface into 4 non-orthogonal triangle faces[ai] comprising a single regular tetrahedral cell: a degenerate 5-cell.[ak] There are 24 ways you can do this, so there are 24 such tetrahedra inscribed in the 24-cell. They overlap with each other, and all their element sets intersect: they share vertex count, edge length, face area, and cell volume, but have no 4-content to share.

#### Three tetrahedral constructions

The 24-cell can be constructed radially from 96 equilateral triangles of edge length 1 which meet at the center of the polytope, each contributing two radii and an edge.[c] They form 96 1 tetrahedra, all sharing the 25th central apex vertex.

The 24-cell can be constructed from 48 equilateral triangles of edge length 2. They form 48 2 tetrahedra (the cells of the three 16-cells), centered at the 24 mid-radii of the 24-cell.

The 24-cell can be constructed from 32 equilateral triangles of edge length 3 centered at the 25th central apex vertex.[k] The edges of these triangles form 96 other equilateral triangles centered at the 24 mid-radii of the 24-cell.[m] These form the faces of 24 3 tetrahedra (the degenerate 5-cells), centered at the 25th central apex vertex.[p]

#### Relationships among interior polytopes

The 24-cell, three tesseracts, three 16-cells and 24 degenerate 5-cells are deeply entwined around their common center. The tesseracts are inscribed in the 24-cell such that their vertices and edges lie on the surface of the 24-cell (they are elements of the 24-cell), but their square faces and cubical cells lie inside the 24-cell (they are not elements of the 24-cell). The 16-cells are inscribed in the 24-cell such that only their vertices lie on the surface: their edges, triangular faces, and tetrahedral cells lie inside the 24-cell. The interior[al] 16-cell edges have length 2. The 5-cells are inscribed in the 24-cell such that only four of their five vertices lie on the surface: their fifth vertex is the center of the 24-cell, and all their edges, triangular faces and tetrahedral cells lie inside the 24-cell.

Kepler's drawing of tetrahedra inscribed in the cube.[32]

The 16-cells are also inscribed in the tesseracts: their 2 edges are the face diagonals of the tesseract, and their 8 vertices occupy every other vertex of the tesseract. Each tesseract has two 16-cells inscribed in it (occupying the opposite vertices and face diagonals), so each 16-cell is inscribed in two of the three 8-cells. This is reminiscent of the way, in 3 dimensions, two tetrahedra can be inscribed in a cube, as discovered by Kepler.[32] In fact it is the exact dimensional analogy (the demihypercubes), and the 48 tetrahedral cells are inscribed in the 24 cubical cells in just that way.[33]

The degenerate 5-cells are each both a regular tetrahedron and an irregular 5-cell. Their 6 long 3 edges are each also a long diagonal of a cube (in two different tesseracts). Their 4 short 1 edges are each also a long radius of the 24-cell (and therefore also a long radius of two different tesseracts). Each 3 tetrahedral face triangle[ai] has one edge entirely in each tesseract, and one vertex in each 16-cell. Only one of the tetrahedral vertices is one end of a coordinate system axis;[am] thus each tetrahedron is spindled on just one of the four coordinate axes.[an]

The 24-cell encloses the three tesseracts within its envelope of octahedral facets, leaving 4-dimensional space in some places between its envelope and each tesseract's envelope of cubes. Each tesseract encloses two of the three 16-cells, leaving 4-dimensional space in some places between its envelope and each 16-cell's envelope of tetrahedra. Thus there are measurable[34] 4-dimensional interstices[ao] between the 24-cell, 8-cell, 16-cell and 5-cell envelopes.[ap] The shapes filling these gaps are 4-pyramids[aq], alluded to above.

#### Boundary cells

Despite the 4-dimensional interstices between 24-cell, 8-cell, 16-cell and 5-cell envelopes, their 3-dimensional volumes overlap. The different envelopes are separated in some places, and in contact in other places (where no 4-pyramid lies between them). Where they are in contact, they merge and share cell volume: they are the same 3-membrane in those places, not two separate but adjacent 3-dimensional layers. Because there are a total of 31 envelopes, there are places where several envelopes come together and merge volume, and also places where envelopes interpenetrate (cross from inside to outside each other).

Some interior features lie inside the (outer) boundary envelope of the 24-cell itself: each octahedral cell is bisected by three perpendicular squares (one from each of the tesseracts), and the diagonals of those squares (which cross each other perpendicularly at the center of the octahedron) are 16-cell edges (one from each 16-cell). Each square bisects an octahedron into two square pyramids, and also bonds two adjacent cubic cells of a tesseract together as their common face.

As we saw above, 16-cell 2 tetrahedral cells are inscribed in tesseract 1 cubic cells, sharing the same volume. 24-cell 1 octahedral cells overlap their volume with 1 cubic cells: they are bisected by a square face into two square pyramids,[35] the apexes of which also lie at a vertex of a cube.[ar] The octahedra share volume not only with the cubes, but with the tetrahedra inscribed in them; thus the 24-cell, tesseracts, and 16-cells all share some boundary volume.[as]

### Configuration

This configuration matrix[36] represents the 24-cell. The rows and columns correspond to vertices, edges, faces, and cells. The diagonal numbers say how many of each element occur in the whole 24-cell. The non-diagonal numbers say how many of the column's element occur in or at the row's element.

${\displaystyle {\begin{bmatrix}{\begin{matrix}24&8&12&6\\2&96&3&3\\3&3&96&2\\6&12&8&24\end{matrix}}\end{bmatrix}}}$

Since the 24-cell is self-dual, its matrix is identical to its 180 degree rotation.

The tessellation of 4-dimensional Euclidean space by regular 24-cells exists; it is called the 24-cell honeycomb. Each 24-cell of this tessellation has 24 neighbors with which it shares an octahedron. It also has 24 other neighbors with which it shares only a single vertex. Eight 24-cells meet at any given vertex in this tessellation. It is one of only three regular tessellations of R4. The unit balls inscribed in the 24-cells of this tessellation give rise to the densest known lattice packing of hyperspheres in 4 dimensions. The vertex configuration of the 24-cell has also been shown to give the highest possible kissing number in 4 dimensions.

The dual tessellation of the 24-cell honeycomb {3,4,3,3} is the 16-cell honeycomb {3,3,4,3}. The third regular tessellation of four dimensional space is the tesseractic honeycomb {4,3,3,4}, whose vertices can be described by 4-integer Cartesian coordinates. The congruent relationships among these three tessellations can be helpful in visualizing the 24-cell, in particular the radial equilateral symmetry which it shares with the tesseract.[c]

A honeycomb of unit-edge-length 24-cells may be overlaid on a honeycomb of unit-edge-length tesseracts such that every vertex of a tesseract (every 4-integer coordinate) is also the vertex of a 24-cell (and tesseract edges are also 24-cell edges), and every center of a 24-cell is also the center of a tesseract. The 24-cells are twice as large as the tesseracts by 4-dimensional content (hypervolume), so overall there are two tesseracts for every 24-cell, only half of which are inscribed in a 24-cell. If those tesseracts are colored black, and their adjacent tesseracts (with which they share a cubical facet) are colored red, a 4-dimensional checkerboard results.[37] Of the 24 center-to-vertex radii[at] of each 24-cell, 16 are also the radii of a black tesseract inscribed in the 24-cell. The other 8 radii extend outside the black tesseract (through the centers of its cubical facets) to the centers of the 8 adjacent red tesseracts. Thus the 24-cell honeycomb and the tesseractic honeycomb coincide in a special way: 8 of the 24 vertices of each 24-cell do not occur at a vertex of a tesseract (they occur at the center of a tesseract instead). Each black tesseract is cut from a 24-cell by truncating it at these 8 vertices, slicing off 8 cubic pyramids (as in reversing Gosset's construction,[25] but instead of being removed the pyramids are simply colored red and left in place). Eight 24-cells meet at the center of each red tesseract: each one meets its opposite at that shared vertex, and the six others at a shared octahedral cell.

The red tesseracts are filled cells (they contain a central vertex and radii); the black tesseracts are empty cells. The vertex set of this union of two honeycombs includes the vertices of all the 24-cells and tesseracts, plus the centers of the red tesseracts. Adding the 24-cell centers (which are also the black tesseract centers) to this honeycomb yields a 16-cell honeycomb, the vertex set of which includes all the vertices and centers of all the 24-cells and tesseracts. The formerly empty centers of adjacent 24-cells become the opposite vertices of a unit-edge-length 16-cell. 24 half-16-cells (octahedral pyramids) meet at each formerly empty center to fill each 24-cell, and their octahedral bases are the 6-vertex octahedral facets of the 24-cell (shared with an adjacent 24-cell).

### Rotations

There are three distinct orientations of the tesseractic honeycomb which could be made to coincide with the 24-cell honeycomb in this manner, depending on which of the 24-cell's three disjoint sets of 8 orthogonal vertices (which set of 4 perpendicular axes) was chosen to align it, just as three tesseracts can be inscribed in the 24-cell, rotated with respect to each other. The distance from one of these orientations to another is an isoclinic rotation through 45 degrees (a double rotation of 45 degrees in each of two orthogonal axes planes, around a single fixed point).[av]

## Notes

1. The pentagon {5}, the dodecahedron {5, 3}, the 600-cell {3,3,5} and the 120-cell {5,3,3}. In other words, the 24-cell possesses all of the triangular and square features that exist in four dimensions, and none of the pentagonal features.
2. The convex regular 4-polytopes can be ordered by size as a measure of 4-dimensional content (hypervolume) for the same radius. Each greater polytope in the sequence is rounder than its predecessor, enclosing more content[19] within the same radius. The 4-simplex (5-cell) is the limit smallest case, and the 120-cell is the largest. Complexity (as measured by comparing configuration matrices or simply the number of vertices) follows the same ordering. This provides an alternative numerical naming scheme for regular polytopes in which the 24-cell is the 4-24-polytope: fourth in the ascending sequence that runs from 4-5-polytope to 4-600-polytope.
3. The long radius (center to vertex) of the 24-cell is equal to its edge length; thus its long diameter (vertex to opposite vertex) is 2 edge lengths. Only a few polytopes have this property, including the four-dimensional 24-cell and tesseract, the three-dimensional cuboctahedron, and the two-dimensional hexagon. (The cuboctahedron is the equatorial cross section of the 24-cell, and the hexagon is the equatorial cross section of the cuboctahedron.) Radially equilateral polytopes are those which can be constructed, with their long radii, from equilateral triangles which meet at the center of the polytope, each contributing two radii and an edge.
4. The edge length will always be different unless predecessor and successor are both radially equilateral, i.e. their edge length is the same as their radius (so both are preserved). Since radially equilateral polytopes[c] are rare, it seems that the only such construction (in any dimension) is from the 8-cell to the 24-cell, making the 24-cell the unique regular polytope (in any dimension) which has the same edge length as its predecessor of the same radius.
5. Up to 6 planes can be mutually orthogonal in 4 dimensions. 3 dimensional space accommodates only 3 perpendicular axes and 3 perpendicular planes through a single point. In 4 dimensional space we may have 4 perpendicular axes and 6 perpendicular planes through a point (for the same reason that the tetrahedron has 6 edges, not 4): there are 6 ways to take 4 dimensions 2 at a time. Three such pairs (perpendicular planes) meet at each vertex (for the same reason that three edges of the tetrahedron meet at each vertex).
6. The edges of the squares are aligned with the grid lines of this coordinate system. For example:
(  0,–1,  1,  0)   (  0,  1,  1,  0)
(  0,–1,–1,  0)   (  0,  1,–1,  0)

is the square in the xy plane. The squares are not made of actual edges of the 24-cell; they are merely invisible configurations of four of the 24-cell's vertices, not visible 24-cell features.
7. Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) they can intersect in a single point: and they must, if and only if they are perpendicular; this is the surprising, counterintuitive thing about how planes intersect in 4-space.
8. The edges of the squares are not aligned with the grid lines of this coordinate system. The squares do lie in the orthogonal planes of the coordinate system, but their edges are the 2 diagonals of the unit-edge-length squares of this coordinate lattice. For example:
(  0,  0,  1,  0)
(  0,–1,  0,  0)   (  0,  1,  0,  0)
(  0,  0,–1,  0)

is the square in the xy plane. The squares are not made of actual unit-length edges of the 24-cell; their edges are chords of length 2.
9. The perpendicular hexagons are inclined (tilted) with respect to the coordinate system's orthogonal planes (containing the perpendicular 2 squares[h] with integer coordinate vertices). Each hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of integer coordinate vertices (one of the four coordinate axes), and two opposite pairs of half-integer coordinate vertices (not coordinate axes). For example:
(  0,  0,  1,  0)
(  1/2,–1/2,  1/2,–1/2)   (  1/2,  1/2,  1/2,  1/2)
(–1/2,–1/2,–1/2,–1/2)   (–1/2,  1/2,–1/2,  1/2)
(  0,  0,–1,  0)
is the hexagon on the y axis. Unlike the 2 squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.
10. It is of course difficult to visualize four hexagonal planes that are all perpendicular to each other. One can see them in the cuboctahedron (a projection of the 24-cell into 3-dimensions), where they appear to be at 60 degrees to each other. In the 3 dimensional projection two of 4 non-orthogonal hexagons appear to intersect at each of 12 vertices, but these are actually 16 hexagons and 24 vertices. In 4 dimensions, 4 non-orthogonal hexagons do intersect at each vertex, but also four orthogonal hexagons intersect only at their common center, such that each one of them passes through a disjoint set of 6 of the 24 vertices.
11. These triangles lie in the same orthogonal planes containing the hexagons;[i] two triangles of edge length 3 are inscribed in each hexagon. For example:
(  0,  0,  1,  0)

(  1/2,–1/2,  1/2,–1/2)   (  1/2,  1/2,  1/2,  1/2)

(–1/2,–1/2,–1/2,–1/2)   (–1/2,  1/2,–1/2,  1/2)

(  0,  0,–1,  0)

are the two opposing central triangles on the y axis. Unlike the hexagons, the 3 triangles are not made of actual 24-cell edges, so they are invisible features of the 24-cell, like the 2 squares.
12. These triangles' edges do not lie in the orthogonal planes of the coordinate system; they are the diagonals of the cubical cells of the coordinate lattice, of length 3.
13. Each of the 96 triangles has edge length 3 and height 3/2, and defines a plane which sections the 24-cell through 3 vertices 1/2 unit-length below a 4th vertex, and 1/2 unit-length above the center, measured from the center of the triangle, which is on a 24-cell diameter joining two opposite vertices; the triangular plane is orthogonal to the diameter line. Each plane contains only one 3 triangle (unlike the central hexagonal planes with their two opposing 3 triangles).[k] The 96 3 triangles are inclined both with respect to the coordinate system's 6 orthogonal planes (the 6 perpendicular 2 squares)[h] and with respect to the 8 3 triangles inscribed in 4 orthogonal planes (the 4 perpendicular hexagons).[i] Each 3 triangle contains one vertex from a square, and two from different hexagons. For example:
(  0,  0,  1,  0)
(  1/2,–1/2,  1/2,  1/2)   (–1/2,  1/2,  1/2,  1/2)

(  1/2,–1/2,–1/2,–1/2)   (–1/2,  1/2,–1/2,–1/2)
(  0,  0,–1,  0)
are the two parallel triangles orthogonal to the y axis. From this perspective, redshifted points are farther from your viewpoint, which is below center looking up.
14. So that the triangles have edge length 3.
15. In 3 dimensions, the coordinates:
${\displaystyle (\pm 1,\pm 1,\pm 1)\in \mathbb {R} ^{3}}$
are the vertices of a pair of dual regular tetrahedra inscribed in a cube.
16. The 96 triangles in 48 parallel pairs are the faces of 24 inscribed tetrahedra in opposing pairs. Each of the 12 diameters of the 24-cell runs from a tetrahedron vertex, through the center of its opposite face, to a vertex of a dual tetrahedron facing the opposite direction.
17. They surround the vertex (in the 3-dimensional space of the 24-cell's boundary surface) the way a cube's 8 corners surround its center. (The vertex figure of the 24-cell is a cube.)
18. They surround the vertex in 3-dimensional space the way an octahedron's 6 corners surround its center.
19. Interior features are not considered elements of the polytope. For example, the center of a 24-cell is a noteworthy feature (as are its long radii), but these interior features do not count as elements in its configuration matrix, which counts only surface features. Interior features are not rendered in most projective illustrations and diagrams of polytopes (they are normally invisible). In illustrations showing interior features, we always draw interior edges as dashed lines, to distinguish them from surface edges.
20. The central vertex is a canonical apex because it is one edge length equidistant from the ordinary vertices in the 4th dimension, as the apex of a canonical pyramid is one edge length equidistant from its other vertices.
21. If the Pythagorean distance between any two vertices is 1, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is 2, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90o bend in it as the path through the center). If their Pythagorean distance is 3, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60o bend, or as a straight line with one 60o bend in it through the center). Finally, if their Pythagorean distance is 4, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).
22. The vertex figure is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a full size vertex figure. That is what serves the illustrative purpose here.
23. Eight 1 edges converge in 3-dimensional space from the corners of the 24-cell's cubical vertex figure[v] and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: six 1-length segments of an apparently straight line (in the 3-space of the 24-cell's 2-sphere surface, here tessellated by cubes) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Viewed from inside this cubical honeycomb 3-space, the bends in the hexagons are invisible. From outside (viewing the 24-cell in 4-space), the straight lines can be seen to bend in the 4th dimension at the cube centers.
24. Six 2 chords converge in 3-space from the face centers of the 24-cell's cubical vertex figure[v] and meet at its center (the vertex), where they form 3 straight lines which cross there perpendicularly. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell, and eight 1 edges converge from there, but let us ignore them now, since 7 straight lines crossing at the center is confusing to visualize all at once. Each of the six 2 chords runs from this cube's center (the vertex) straight through a face center to the center of an adjacent (face-bonded) cube, which is another vertex of the 24-cell: not a nearest vertex (at the cube corners), but one located 90° away in a second concentric shell of six 2-distant vertices that surrounds the first shell of eight 1-distant vertices. The face-center through which the 2 chord passes is the mid-point of the 2 chord, so it lies inside the 24-cell, not on its surface.
25. One can cut the 24-cell into two equal parts through 6 vertices (in any hexagonal great circle plane), or through 4 vertices (in any square great circle plane). One can see this in the cuboctahedron (the central hyperplane of the 24-cell), where there are four hexagonal great circles (along the edges) and six square great circles (across the square faces diagonally).
26. Eight 3 chords converge from the corners of the 24-cell's cubical vertex figure[v] and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell, and eight 1 edges also converge from there, and six 2 chords converge from the face centers, but let us ignore them now, since so many straight lines crossing at the center is confusing to visualize all at once. Each of the eight 3 chords runs from this cube's center straight through a nearest vertex (the cube's vertex) to the center of a diagonally adjacent (vertex-bonded) cube, which is another vertex of the 24-cell: one located 120° away in a third concentric shell of eight 3-distant vertices surrounding the second shell of six 2-distant vertices that surrounds the first shell of eight 1-distant vertices.
27. Three dimensional rotations occur around an axis line. Four dimensional rotations may occur around a plane. So in three dimensions we may fold planes around a common line (as when folding a flat net of 6 squares up into a cube), and in four dimensions we may fold cells around a common plane (as when folding a flat net of 8 cubes up into a tesseract). Folding around a square face is just folding around two of its orthogonal edges at the same time; there is not enough space in three dimensions to do this, just as there is not enough space in two dimensions to fold around a line (only enough to fold around a point).
28. Each great circle plane intersects with the other great circle planes to which it is not orthogonal at one 4 diameter of the 24-cell. Thus two non-orthogonal great circles share two vertices (unlike two orthogonal great circles which share no points except their common center). Each face plane intersects with the other face planes of its kind to which it is not orthogonal at their characteristic edge. It may seem paradoxical that the face planes of orthogonally-faced cells (such as cubes) intersect at an edge (as they obviously do), since planes are not supposed to be able to intersect in 4-space (except at a single point) if they are perpendicular. The resolution of this apparent paradox is that the face planes of 4-polytope cells are folded in the 4th dimension around their edges of intersection (as the boundary 3-manifold of the 4-polytope is folded around its face planes[aa]), so if their dihedral angle is 90 degrees in the boundary 3-space, it is some other angle in 4-space, and they are not orthogonal in 4-space.
29. The 600-cell is larger than the 24-cell, and contains the 24-cell as an interior feature.[23] The regular 5-cell is not found in the interior any 4-polytope except the 120-cell,[24] though every 4-polytope can be deconstructed into irregular 5-cells.
30. The only planes through exactly 6 vertices of the 24-cell (not counting the central vertex) are the 16 hexagonal great circles. There are no planes through exactly 5 vertices. There are two kinds of planes through exactly 4 vertices: the 18 2 square great circles, and 72 1 square (tesseract) faces. There is always a plane through any 3 vertices; some of them are actually through exactly 6 (the 32 3 equilateral triangles in the central hexagons), some are actually through exactly 4 (isosceles right triangles with a 3 hypotenuse in the central 2 squares, and isosceles right triangles with a 2 hypotenuse in the 1 squares), but some are through exactly 3: 96 3 equilateral triangle (5-cell) faces that do not lie in a central plane, and 96 2 equilateral triangle (16-cell) faces; 1 2 2 and 2 3 3 isosceles triangles, and 1 2 3 scalene right triangles; and 96 1 equilateral triangle (24-cell) faces.
31. The 24-cell's cubical vertex figure[v] has been truncated to a tetrahedral vertex figure (see Kepler's drawing). The vertex cube has vanished, and now there are only 4 corners of the vertex figure where before there were 8. Four tesseract edges converge from the tetrahedron vertices and meet at its center, where they do not cross (since the tetrahedron does not have opposing vertices).
32. The 24-cell's cubical vertex figure[v] has been truncated to an octahedral vertex figure. The vertex cube has vanished, and now there are only 6 corners of the vertex figure where before there were 8. The 6 2 chords which formerly converged from cube face centers now converge from octahedron vertices; but just as before, they meet at the center where 3 straight lines cross perpendicularly. The octahedron vertices are located 90° away outside the vanished cube, at the new nearest vertices; before truncation those were 24-cell vertices in the second shell of surrounding vertices.
33. Starting at any vertex, find one of the 4 3 triangular face planes[m] centered 1/2 edge-length below it and slice through those 3 vertices, removing the starting vertex and 4 of its 8 nearest neighbors and exposing a 3 triangular face. Repeat for the other three sides of the tetrahedron.
34. The 24-cell's cubical vertex figure[v] has been truncated to a triangular vertex figure. The vertex cube has vanished, and now there are only 3 corners of the vertex figure where before there were 8. Three of the eight 3 chords which formerly converged from cube vertices now converge from triangle vertices; they meet at the triangle's center, where now they do not cross (because the triangle does not have opposing vertices). The triangle vertices are located 120° away outside the vanished cube, at the new nearest vertices; before truncation they were 24-cell vertices in the third shell of surrounding vertices. The three remaining converging 3 chords are now three of the 5-cell's 6 3 edges. The 5-cell has 10 edges, but being irregular, it has only 6 3 edges, three of which meet at this vertex. Its other 4 edges are 1 24-cell radii (because the 5-cell's 5th vertex is the 24-cell center), only one of which is incident to this vertex. Since those radii are perpendicular to the tetrahedral bounding surface of the irregular 5-cell, they converge perpendicularly from the 4th dimension directly at the center of the vertex figure; thus the one 1 edge that meets three 3 edges at this vertex is invisible because it is foreshortened to a point in the 3-dimensional space here.
35. These 3 tetrahedron faces are not the same triangles as the great circle 3 triangles. They are both the same size because they are made from the same 96 3 chords, so they are easily confused, but they lie in different planes, and there are different quantities of them. The 32 great circle triangles[k] lie in 16 hexagonal planes (4 sets of 4 mutually orthogonal planes), all with their centers at the center of the 24-cell. In contrast, 96 3 triangular tetrahedron faces lie in 48 sets of 2 parallel planes, none of which pass through the center of the 24-cell.[m] The three vertices that each face triangle connects do not lie in the same great circle of the 24-cell. Each face plane divides the 24-cell into two unequal parts: there are 5 vertices above the plane, 3 and only 3 vertices in the face plane, and 16 below it (not counting the 25th central vertex 1/2 unit-length below the face center).
36. One way to visualize the n-dimensional hyperplanes is as the n-spaces which can be defined by n + 1 points. A point is the 0-space which is defined by 1 point. A line is the 1-space which is defined by 2 points which are not coincident. A plane is the 2-space which is defined by 3 points which are not colinear (any triangle). In 4-space, a 3-dimensional hyperplane is the 3-space which is defined by 4 points which are not coplanar (any tetrahedron). In 5-space, a 4-dimensional hyperplane is the 4-space which is defined by 5 points which are not cocellular (any 5-cell). These simplex figures divide the hyperplane into two parts (inside and outside the figure), but in addition they divide the universe (the enclosing space) into two parts (above and below the hyperplane). The n points bound a finite simplex figure (from the outside), and they define an infinite hyperplane (from the inside).[30] These two divisions are orthogonal, so the defining simplex divides space into six regions: inside the simplex and in the hyperplane, inside the simplex but above or below the hyperplane, outside the simplex but in the hyperplane, and outside the simplex above or below the hyperplane.
37. The 3 tetrahedron is one cell of a canonical apex 5-cell because it is the base of a canonical tetrahedral pyramid with its apex at the center of the 24-cell. The base surrounds the central apex so its vertices are equidistant from it. Tetrahedral pyramid is another name for a 5-cell. Even though the tetrahedron is regular, the 5-cell is irregular, because its edge length is 3 but its height is 1. Its other four cells are irregular tetrahedra (the sides of the pyramid) which meet at the center. As well as being irregular, the 5-cell is degenerate, because its 5 points are cocellular: the central apex lies exactly in the hyperplane[aj] defined by the other 4 vertices. The 3 tetrahedron is a degenerate 5-cell in exactly the same sense that a 3-dimensional honeycomb is a degenerate 4-polytope: it defines a 3-dimensional hyperplane, but it fails to define a 4-space and it has zero 4-dimensional content, because it is flat in the 4th dimension.[31]
38. The edges of the 16-cells are not visible in projective renderings of the 24-cell; in diagrams which show interior edges, they are drawn as dashed lines. The edges of the inscribed tesseracts are always visible, because they are also edges of the 24-cell (they lie on its surface).
39. Each 3 tetrahedron has one vertex in each of 4 orthogonal hexagons.[i] Thus none of its 3 edges lie in those 4 hexagonal great circle planes; they lie in 6 others.
40. There are two √3 tetrahedra spindled on each of the 12 vertex-to-vertex diameters, one pointing in each direction.
41. The 4-dimensional content of the unit-edge-length tesseract is 1 (by definition). The content of the unit-edge-length 24-cell is 2, so half its content is inside each tesseract, and half is between their envelopes. Each 16-cell (edge length 2) encloses a content of 2/3, leaving 1/3 of an enclosing tesseract between their envelopes.
42. Although the degenerate 5-cell is not wholly enclosed (as a subset of vertices) within any one tesseract or 16-cell, since it has zero 4-dimensional content it is still possible to measure the content between its (4-dimensionally flat) envelope and any other 4-polytope's envelope: it is simply the content of that 4-polytope.
43. Between the 24-cell envelope and the 8-cell envelope, we have the 8 cubic pyramids of Gosset's construction. Between the 8-cell envelope and the 16-cell envelope, we have 16 right tetrahedral pyramids, with their apexes filling the corners of the tesseract.
44. This might appear at first to be angularly impossible, and indeed it would be in a flat space of only three dimensions. If two cubes rest face-to-face in an ordinary 3-dimensional space (e.g. on the surface of a table in an ordinary 3-dimensional room), an octahedron will fit inside them such that four of its six vertices are at the four corners of the square face between the two cubes; but then the other two octahedral vertices will not lie at a cube corner (they will fall within the volume of the two cubes, but not at a cube vertex). In four dimensions, however, the boundary 3-spaces of 4-polytopes are bent. The tesseract's boundary 3-manifold (a tessellation of the 3-sphere by 8 cubes) is folded around its square face planes,[aa] so that the adjacent face-bonded cubes are oriented with respect to each other differently than they would be in an ordinary 3-dimensional room, such that all 6 of the octahedron's vertices lie at the vertex of a cube. This is only possible because the 24-cell's boundary 3-manifold (a tessellation of the 3-sphere by 24 octahedra) is folded differently, around its triangular face planes, i.e. it is folded in different places than the tesseract's 3-manifold. The individual octahedra are not bent at the square cube faces which are their central sections; the individual cubes are not bent either; each 4-polytope's 3-manifold folds only at its own characteristic face planes, independently of the way the other 4-polytope is folded.
45. Consider the three perpendicular 2 long diameters of the octahedral cell. Two of them are the face diagonals of the square face between two cubes; each is a 2 chord that connects two vertices of an 8-cell cube across a square face, connects two vertices of 16-cell tetrahedron (inscribed in the cube), and connects two vertices of a 24-cell octahedron across a square central section. The third perpendicular long diameter of the octahedron does exactly the same (by symmetry); so it also connects two vertices of a cube across a square face (but a face of a different pair of cubes, from one of the other tesseracts in the 24-cell).
46. It is important to visualize the radii only as invisible interior features of the 24-cell (dashed lines), since they are not edges of the honeycomb. Similarly, the center of the 24-cell is empty (not a vertex of the honeycomb).
47. There are (at least) two kinds of correct dimensional analogies: the usual kind between dimension n and dimension n + 1, and the much rarer and less obvious kind between dimension n and dimension n + 2. An example of the latter is that rotations in 4-space may take place around a single point, as do rotations in 2-space. Another is that the surface area of the (n+2)-sphere is exactly 2π r times the volume enclosed by the n-sphere, the most well-known example being that the circumference of a 2-sphere is 2π r times the length of a 0-sphere. Coxeter cites[38] this as an instance in which dimensional analogy can fail us as a method, but it is really our failure to recognize whether a one- or two-dimensional analogy is the appropriate method.
48. Rotations in four dimensions may occur around a plane, as when adjacent cells are folded around their plane of intersection (by analogy to the way adjacent faces are folded around their line of intersection).[aa] But in four dimensions there is yet another way in which rotations can occur, called a double rotation. Double rotations are an emergent phenomenon in the fourth dimension and have no analogy in three dimensions: folding up square faces and folding up cubical cells are both examples of single rotations, the only kind that occur in fewer than four dimensions. In 3-dimensional rotations, the points in a line remain fixed during the rotation, while every other point moves. In 4-dimensional single rotations, the points in a plane remain fixed during the rotation, while every other point moves. In 4-dimensional double rotations, a point remains fixed during rotation, and every other point moves (as in a 2-dimensional rotation!). This is one of several surprising, counter-intuitive things about rotations in 4-space.[au]

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18. Coxeter 1973, Chapter VII: Ordinary Polytopes in Higher Space, p. 118.
19. Coxeter 1973, Table I(ii): The sixteen regular polytopes {p,q,r} in four dimensions, pp. 292-293: [An invaluable table providing all 20 metrics of each 4-polytope.]
20. Coxeter 1973, Table VI (ii): 𝐈𝐈 = {3,4,3}, p. 302: see Result column
21. Coxeter 1973, §8.7. Cartesian Coordinates, p. 156.
22. Coxeter 1973, Table V: The Distribution of Vertices of Four-Dimensional Polytopes in Parallel Solid Sections (§13.1); (i) Sections of {3,4,3} (edge 2) beginning with a vertex; see column a, p. 298.
23. Coxeter 1973, Table VI(iii) II={3,3,5}, p. 303: Faceting 5{3,3,5}[25{3,4,3}]{3,3,5} of the 600-cell reveals 25 regular 24-cells.
24. Coxeter 1973, Table VI(iv) II={5,3,3}, p. 304: Faceting {5,3,3}[120𝛼4]{3,3,5} of the 120-cell reveals 120 regular 5-cells.
25. Coxeter 1973, Gosset, p. 150.
26. Coxeter 1973, §8.2. Cesaro's construction for {3, 4, 3}., p. 148.
27. Coxeter 1973, Table VI(ii) II={3,4,3}, Result column, p. 302.
28. Coxeter 1973, §8.22. see illustrations Fig. 8.2A and Fig 8.2B, pp. 149-150
29. Coxeter 1973, §8.1 The simple truncations of the general regular polytope, pp. 145-146.
30. Coxeter 1973, §7.2., p. 120: "... any n+1 points which do not lie in an (n-1)-space are the vertices of an n-dimensional simplex.... Thus the general simplex may alternatively be defined as a finite region of n-space enclosed by n+1 hyperplanes or (n-1)-spaces."
31. Coxeter 1973, IV Tessellations and Honeycombs, p. 58.
32. Kepler 1619, p. 181.
33. Coxeter 1973, §14.32, p. 269. "For instance, in the case of ${\displaystyle \gamma _{4}[2\beta _{4}]}$ ...."
34. Coxeter 1973, Table I (ii) The sixteen regular polytopes {p, q, r} in four dimensions, pp. 292-293. [An invaluable table providing all 20 metrics of each 4-polytope.]
35. Coxeter 1973, p. 150: "Thus the 24 cells of the {3, 4, 3} are dipyramids based on the 24 squares of the ${\displaystyle \gamma _{4}}$ . (Their centres are the mid-points of the 24 edges of the ${\displaystyle \beta _{4}}$ .)"
36. Coxeter 1973, §1.8. Configurations, p. 12.
37. Coxeter 1973, p. 156: "...the chess-board has an n-dimensional analogue."
38. Coxeter 1973, §7.1. Dimensional Analogy, p. 119: "For instance, seeing that the circumference of a circle is 2π r, while the surface of a sphere is 4π r 2, ... it is unlikely that the use of analogy, unaided by computation, would ever lead us to the correct expression, 2π 2r 3."