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Tessellations of the regular figures lead to the symmetries of division of angles that the base of numeration should be chosen to represent compactly. Not all regular figures tessellate the Euclidean non-curved spaces of area and volume and capacities of higher dimensions.
Whereas other mathematicians may mean by a circle its circumference, and by a sphere its surface, here rather a circle would be the circumference and the disk, and a sphere would be the surface and the volume, the radius and centre, and all the features of the object. Likewise, by a figure is here meant the enclosed space and its non-curved boundary.
The regular four-dimensional figures are bound by regular three-dimensional figures of the same kind, out of the five Platonic solids.
Lemma: The demonstration that there are no more than five regular solids
I have read that this proof can be credited to Euclid at the end of his books on the Elements of geometry.
The regular three-dimensional Platonic solid figures are bound by regular two-dimensional figures, the regular polygons as faces.
To form a regular three-dimensional solid, a number of regular polygons are joined at each vertex.
The sum of the angles of the polygons at a vertex cannot exceed the angle of a full circular turn or perigon, which is \(2\pi\) radians or two-and-a-half square dozen degrees, or else the polygons would tile a non-flat "concave" curved non-Euclidean hyperbolic surface.
If the sum of the angles of the polygons at the vertex equals a full turn, the polygons tile the flat Euclidean plane, and do not fold about a third dimension or form solids.
There must be at least three polygons at a vertex in order for them to fold about a third dimension without gaps between their edges. If there were only two faces, they would enclose no volume or their hinge would not be rigid.
1. Of the Triangle
The simplest regular polygon having a face is the equilateral triangle.
The sum of the three angles of a plane flat Euclidean triangle is the angle of a half turn.
The angles at the three vertices of the equilateral triangle are all equal to each other, therefore they are a third of a half turn, and a sixth of a full turn.
The sum of three angles of three equilateral triangles is less than a full turn, therefore they do not tile a plane and can fold about the third dimension. (They happen to form, by closure with a fourth triangle, the regular tetrahedron, the simplest one of the Platonic solids.)
The angles at the apices of four equilateral triangles sum to less than a full turn. Therefore, they can be folded about the third dimension, as the triangular faces of a square-based pyramid. (The resulting figure, with closure by another four triangles in the form of second such pyramid, forms the regular octahedron, another of the five Platonic solids.)
The five angles at the vertices of five equilateral triangles joined together at a vertex sum to less than a full turn. Therefore, the five triangles can be folded about the third dimension. (The resulting figure with closure by continuing the pattern of five triangles to each vertex forms the Platonic solid having twenty faces, called the icosahedron.)
The sum of six vertex angles of equilateral triangles joined at a point equals a full turn and tiles the plane.
More than six equilateral triangles exceed the perigon and tile a hyperbolic surface.
2. Of the Square
The next regular polygon is the square. The angle at its corners is a right angle or quarter of a perigon, in degrees 90°, and in radians π/2.
The angles at the corners of three squares joined at a vertex would be less than a full turn, therefore they can fold about the third dimension. (The figure when closed with its duplication forms the cube.)
The angles at the corners of four squares joined at a vertex would sum to the full plane angle and would not fold about the third dimension.
3. Of the Pentagon
The angle at the vertex of the regular pentagon is \(3\pi/5\) radians, which is less than a third of the perigon. Therefore, three pentagons fit into the turn with room to spare, and can fold about the third dimension without gaps between the edges. (This procedure repeated at each vertex forms the Platonic regular solid of twelve pentagonal faces)
4. Of the Hexagon
The angle at vertices of the regular hexagon is a third of the turn, so three hexagons at a vertex tile the flat plane and do not fold about the third dimension.
5. Of the Heptagon
The angle at vertices of the heptagon are larger than a third of a turn, and therefore it is not possible for at least three heptagons to join at a vertex and fold about the third dimension convexly.
The same applies to all regular polygons of more edges.
Thus there can be no more than five regular solids. There are indeed exactly as many as five, which are the Platonic solids.
Of Figures in Four Dimensions
For polyhedra to fold together about the fourth dimension, the sum of the solid angles at their vertices where they join at a vertex must not equal or exceed the full solid angle of the sphere, or else the solids would tessellate the three-dimensional space flatly or a hyperbolic space curvedly.
Proposition: There must be at least four solids joined at a vertex.
Therefore, the solid angle at the vertex of the polyhedron must not exceed a quarter of the full solid angle of the sphere in three-dimensional space. Hence, the solid angle at the vertex of the polyhedron must be less than \(\pi\) steradians, a quarter of the total space solid angle \(4\pi\) steradians.
Proposition: There must be at least three polyhedra joined at each edge where they fold.
Thus, the dihedral angle between any two faces of the polyhedron cannot exceed and must be less than a third of a full turn. Further, the sum of the dihedral angles at the edges where the polyhedra are joined and fold cannot equal or exceed the full turn angle of a plane, or else the polyhedra would brick the lower space of three dimensions or a curved hyperbolic space.
Consequently, the solid angles at the vertices and dihedral angles at the edges between the faces of the five regular polyhedra were calculated. The relative solid angle is determined as the proportion of the area taken up from the surface of a sphere centred at the vertex by a ring of edges projected from the centre onto it. Dihedral angles were calculated first, followed by solid angles.
1. Bounded by Tetrahedra
The solid angle at the vertex of the regular tetrahedron was calculated to divide into the maximum spherical angle between a dozen plus ten times and a dozen plus eleven times. Therefore, the number of tetrahedra joined at a junction and folding about the fourth dimension cannot exceed a dozen plus eleven, but there is plenty of room for fewer to do so.
The dihedral angle between the faces of the regular tetrahedron was calculated to divide into the full turn between five and six times. Therefore, fewer than six tetrahedra could join at an edge and fold there about the fourth dimension.
Thus, by the dihedral angles:
Three tetrahedra may join at an edge and fold about the fourth dimension. (This allows the pentatope.)
Four tetrahedra may fold together about the fourth dimension. (This allows the square four cell.)
Five tetrahedra may be joined and fold. (This permits the four square dozen plus two dozen cell.)
2. Bound by Octahedra
The solid angle at the vertex of the octahedron was found by dissecting the projection of the square into two spherical triangles, the equal areas of which were calculated and added together. The solid angle of the octahedron was calculated to divide into the full spherical angle between nine and ten times, which is more than four times, so they may be joined to fold about the fourth dimension not more than nine times by this constraint.
The dihedral angle at an edge of the octahedron is less than a third of a turn, which is small enough for the necessary three polyhedra per edge, and goes into a turn between three and four times, meaning that not more than three octahedra can fold convexly. (The three octahedra per edge allows the double dozen cell.)
3. Bound by Icosahedra
The solid angle of this solid goes into the full space solid angle between four and five times, and even though it is smaller than the solid angle at the vertex of the regular solid of twelve pentagonal faces, the dihedral angle at edges between faces is larger than a third of a turn, so it is not possible for the necessary at least three solids to join at an edge to fold about the fourth dimension.
4. Bound by Cubes
The solid angle of a cube is an eighth of the space, and this is less than a quarter, so folding may occur.
The dihedral angle is a quarter turn, so not as many as four cubes can join at an edge or they would brick the lower three-dimensional space and not fold about the fourth dimension. Therefore, there can be only three cubes at an edge. (This allows the tesseract.)
5. Bound by Dodecahedra.
The solid angle at a vertex of the regular solid of twelve pentagonal faces is the largest vertex solid angle of the five regular solids but is just less than a quarter of the full space solid angle, so four of them may be joined at a vertex and fold about the fourth dimension.
The dihedral angle does not exceed a third of a turn and divides into a perigon just more than three times, so their joining and folding about the fourth dimension is allowed. (The result is the twelfty cell.)
Thus, no more than six regular four-dimensional figures are possible.
By slopes, the dihedral angles between faces, whether connected at edges or vertices or disconnected, in the Platonic solids were calculated about the year 2001, after all the distances between vertices. Solid angles were calculated soon afterwards using spherical excess, the formula of which was known before it was derived. An outline of this argument in thought is probably at least ten years old. Graphs for the pentatope and tesseract were known, and much of the structure of the pentatope was calculated. However, graphs for the more complicated four-dimensional figures were not worked out.
Whereas other mathematicians may mean by a circle its circumference, and by a sphere its surface, here rather a circle would be the circumference and the disk, and a sphere would be the surface and the volume, the radius and centre, and all the features of the object. Likewise, by a figure is here meant the enclosed space and its non-curved boundary.
The regular four-dimensional figures are bound by regular three-dimensional figures of the same kind, out of the five Platonic solids.
Lemma: The demonstration that there are no more than five regular solids
I have read that this proof can be credited to Euclid at the end of his books on the Elements of geometry.
The regular three-dimensional Platonic solid figures are bound by regular two-dimensional figures, the regular polygons as faces.
To form a regular three-dimensional solid, a number of regular polygons are joined at each vertex.
The sum of the angles of the polygons at a vertex cannot exceed the angle of a full circular turn or perigon, which is \(2\pi\) radians or two-and-a-half square dozen degrees, or else the polygons would tile a non-flat "concave" curved non-Euclidean hyperbolic surface.
If the sum of the angles of the polygons at the vertex equals a full turn, the polygons tile the flat Euclidean plane, and do not fold about a third dimension or form solids.
There must be at least three polygons at a vertex in order for them to fold about a third dimension without gaps between their edges. If there were only two faces, they would enclose no volume or their hinge would not be rigid.
1. Of the Triangle
The simplest regular polygon having a face is the equilateral triangle.
The sum of the three angles of a plane flat Euclidean triangle is the angle of a half turn.
The angles at the three vertices of the equilateral triangle are all equal to each other, therefore they are a third of a half turn, and a sixth of a full turn.
The sum of three angles of three equilateral triangles is less than a full turn, therefore they do not tile a plane and can fold about the third dimension. (They happen to form, by closure with a fourth triangle, the regular tetrahedron, the simplest one of the Platonic solids.)
The angles at the apices of four equilateral triangles sum to less than a full turn. Therefore, they can be folded about the third dimension, as the triangular faces of a square-based pyramid. (The resulting figure, with closure by another four triangles in the form of second such pyramid, forms the regular octahedron, another of the five Platonic solids.)
The five angles at the vertices of five equilateral triangles joined together at a vertex sum to less than a full turn. Therefore, the five triangles can be folded about the third dimension. (The resulting figure with closure by continuing the pattern of five triangles to each vertex forms the Platonic solid having twenty faces, called the icosahedron.)
The sum of six vertex angles of equilateral triangles joined at a point equals a full turn and tiles the plane.
More than six equilateral triangles exceed the perigon and tile a hyperbolic surface.
2. Of the Square
The next regular polygon is the square. The angle at its corners is a right angle or quarter of a perigon, in degrees 90°, and in radians π/2.
The angles at the corners of three squares joined at a vertex would be less than a full turn, therefore they can fold about the third dimension. (The figure when closed with its duplication forms the cube.)
The angles at the corners of four squares joined at a vertex would sum to the full plane angle and would not fold about the third dimension.
3. Of the Pentagon
The angle at the vertex of the regular pentagon is \(3\pi/5\) radians, which is less than a third of the perigon. Therefore, three pentagons fit into the turn with room to spare, and can fold about the third dimension without gaps between the edges. (This procedure repeated at each vertex forms the Platonic regular solid of twelve pentagonal faces)
4. Of the Hexagon
The angle at vertices of the regular hexagon is a third of the turn, so three hexagons at a vertex tile the flat plane and do not fold about the third dimension.
5. Of the Heptagon
The angle at vertices of the heptagon are larger than a third of a turn, and therefore it is not possible for at least three heptagons to join at a vertex and fold about the third dimension convexly.
The same applies to all regular polygons of more edges.
Thus there can be no more than five regular solids. There are indeed exactly as many as five, which are the Platonic solids.
Of Figures in Four Dimensions
For polyhedra to fold together about the fourth dimension, the sum of the solid angles at their vertices where they join at a vertex must not equal or exceed the full solid angle of the sphere, or else the solids would tessellate the three-dimensional space flatly or a hyperbolic space curvedly.
Proposition: There must be at least four solids joined at a vertex.
Therefore, the solid angle at the vertex of the polyhedron must not exceed a quarter of the full solid angle of the sphere in three-dimensional space. Hence, the solid angle at the vertex of the polyhedron must be less than \(\pi\) steradians, a quarter of the total space solid angle \(4\pi\) steradians.
Proposition: There must be at least three polyhedra joined at each edge where they fold.
Thus, the dihedral angle between any two faces of the polyhedron cannot exceed and must be less than a third of a full turn. Further, the sum of the dihedral angles at the edges where the polyhedra are joined and fold cannot equal or exceed the full turn angle of a plane, or else the polyhedra would brick the lower space of three dimensions or a curved hyperbolic space.
Consequently, the solid angles at the vertices and dihedral angles at the edges between the faces of the five regular polyhedra were calculated. The relative solid angle is determined as the proportion of the area taken up from the surface of a sphere centred at the vertex by a ring of edges projected from the centre onto it. Dihedral angles were calculated first, followed by solid angles.
1. Bounded by Tetrahedra
The solid angle at the vertex of the regular tetrahedron was calculated to divide into the maximum spherical angle between a dozen plus ten times and a dozen plus eleven times. Therefore, the number of tetrahedra joined at a junction and folding about the fourth dimension cannot exceed a dozen plus eleven, but there is plenty of room for fewer to do so.
The dihedral angle between the faces of the regular tetrahedron was calculated to divide into the full turn between five and six times. Therefore, fewer than six tetrahedra could join at an edge and fold there about the fourth dimension.
Thus, by the dihedral angles:
Three tetrahedra may join at an edge and fold about the fourth dimension. (This allows the pentatope.)
Four tetrahedra may fold together about the fourth dimension. (This allows the square four cell.)
Five tetrahedra may be joined and fold. (This permits the four square dozen plus two dozen cell.)
2. Bound by Octahedra
The solid angle at the vertex of the octahedron was found by dissecting the projection of the square into two spherical triangles, the equal areas of which were calculated and added together. The solid angle of the octahedron was calculated to divide into the full spherical angle between nine and ten times, which is more than four times, so they may be joined to fold about the fourth dimension not more than nine times by this constraint.
The dihedral angle at an edge of the octahedron is less than a third of a turn, which is small enough for the necessary three polyhedra per edge, and goes into a turn between three and four times, meaning that not more than three octahedra can fold convexly. (The three octahedra per edge allows the double dozen cell.)
3. Bound by Icosahedra
The solid angle of this solid goes into the full space solid angle between four and five times, and even though it is smaller than the solid angle at the vertex of the regular solid of twelve pentagonal faces, the dihedral angle at edges between faces is larger than a third of a turn, so it is not possible for the necessary at least three solids to join at an edge to fold about the fourth dimension.
4. Bound by Cubes
The solid angle of a cube is an eighth of the space, and this is less than a quarter, so folding may occur.
The dihedral angle is a quarter turn, so not as many as four cubes can join at an edge or they would brick the lower three-dimensional space and not fold about the fourth dimension. Therefore, there can be only three cubes at an edge. (This allows the tesseract.)
5. Bound by Dodecahedra.
The solid angle at a vertex of the regular solid of twelve pentagonal faces is the largest vertex solid angle of the five regular solids but is just less than a quarter of the full space solid angle, so four of them may be joined at a vertex and fold about the fourth dimension.
The dihedral angle does not exceed a third of a turn and divides into a perigon just more than three times, so their joining and folding about the fourth dimension is allowed. (The result is the twelfty cell.)
Thus, no more than six regular four-dimensional figures are possible.
By slopes, the dihedral angles between faces, whether connected at edges or vertices or disconnected, in the Platonic solids were calculated about the year 2001, after all the distances between vertices. Solid angles were calculated soon afterwards using spherical excess, the formula of which was known before it was derived. An outline of this argument in thought is probably at least ten years old. Graphs for the pentatope and tesseract were known, and much of the structure of the pentatope was calculated. However, graphs for the more complicated four-dimensional figures were not worked out.
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