Geometry of Arbitrary Dimension

Axiomatic Geometry

Primitives and Dimension

Geometry is an attempt to describe our surroundings using axiomatic approach. Some shapes were chosen as primitives and some statements about them, considered intuitively obvious, as axioms. What are these primitives and these axioms depends on the type of surrounding you are trying to describe. There are some common threads though. Primitives form a numeric progression: there are 0-primitives, 1-primitives, and so on. d-primitives are also called primitives of dimension d. 0-primitives are typically called Points. However, in Space-Time geometry a Point is a 1-primitive and a 0-primitive (a location of a Point at a particular time) is called an Event. 1-primitives are typically called lines. However 1-primitives on a Sphere are called Great Circles. The entire surrounding is also assigned a dimension. It is a number d such that the dimensions of the primitives it contains are between 0 and d-1. For example, Space-Time can be partitioned by grouping together all events which happen simultaneously. But events which happen at the same time is nothing more than a 3-dimensional Space. So Space-Time is 4-dimensional. If a p-primitive lives in a d-dimensional surrounding then the number d-p is called its codimension. Primitives may or may not intersect but when they do, the intersection is also a primitive. In particular, for any pair of primitives, the intersection of all primitives containing them both (if such exist) is called the span of that pair. If such do not not exist, we say that the pair spans the entire surrounding.

If a pair of primitives spans the surrounding and intersects
then the sum of their codimensions is equal to the codimension of the intersection.

The situation where primitives U, V span the surrounding and their intersection is a 0-primitive is often described by saying the the surrounding is a decomposition U⊕V. If, in addition, U and V are orthogonal, we say that the surrounding is an orthogonal decomposition U⊥V.

Projective Plane

Note that 2 great circles intersect at a pair of antipode points. So 0-primitives of a Spherical geometry are such antipode pairs rather than individual points. The geometry of antipode pairs has many similarities to an ordinary geometry so we simply call these 0-primitives - Points and rename the surrounding calling it the Projective Plane ℙ₂. Removing a line cuts a plane into 2 disconnected parts, called half-planes. Removing a line from ℙ₂ leaves just one connected part since an antipode always lives on both sides of the divide. That connected part, as far as the primitives go, behaves like an ordinary plane. But in contrast to what happens on an ordinary plane, the distance between any pair of points is bounded by number (namely half of the circumference of a great circle). Hence when a shape is transplanted from its natural spherical surrounding to a flat plane it appears distorted. With these precautions in mind we can summarize that a ℙ₂ is just an ordinary plane with an extra line added (at infinity). That leads to a phrase: 2 parallel lines intersect at infinity.

Vocabulary of Higher Dimensions

When trying to come up with a name which extrapolates something familiar to an arbitrary dimension we often use the prefix Hyper. So a Hypersphere centered at a point is a shape made of points at a fixed distance from the center. A shape is bounded if it can be enclosed by a Hypersphere. A bounded shape has a well defined Hypervolume and its boundary, a Hyperarea. A primitive of codimension 1 is called a Hyperplane. The surroundings that are familiar to us have dimensions at most 4. We can however make valid statements about primitives of any dimension leaving aside the question of which reality they attempt to describe.

Concept of Polytope

A plane polygon (such as a triangle) and of a space Polyhedron (such as a cube) have a Hyperspace generalization called a Polytope. A Polytope is defined to be a shape bound by a finite collection of Hyperplanes. The only Polytopes of dimension 1 are a segment and a half-line. A Polytope which lives in a dimension p can be viewed as an aggregate of Polytopes of dimensions lower than p. Some of these were given names

dimension	name
p-1	Hyperface
p-2	Hyperedge
2	Face
1	Edge
0	Vertex

If P ⊂ V is a Polytope and W is some hyperspace then P⊥W ⊂ V⊥W is also a Polytope. For example if P is a Polygone and W a Line then P⊥W is an infinite column with cross-section P. Every polytope can be represented in a unique way as P⊥W where W is some Hyperspace and P contains at least 1 vertex. More generally, if P ⊂ V and Q ⊂ W are Polytopes then P⊥Q ⊂ V⊥W is a Polytope (P⊥Q = P⊥W∩ V⊥Q ). For example if P is a Polygone and Q a segment then P⊥Q is a Prism with base P and height Q.

Maximal Decomposition Theorem

Every Polytope has a representation of the form

W⊥P₁⊥P₂⊥ … ⊥P_n

of maximal length where W is a Hyperspace and P_i are vertex-polytopes of dimension ≥ 1. This representation is unique up to an ordering of the terms.

A Solid-angle is a polytope with 1 vertex. It can also be viewed as a hyperspherical shape obtained by intersecting the Solid-angle with a unit Hypersphere centered at the vertex of the angle. A magnitude of a Solid-angle may be defined as the hyperarea of the intersection of the angle with the Hypersphere. Any vertex of a Polytope defines a solid angle bound by the Hyperfaces passing through that vertex. A Special vertex of a Polytope is one for which the solid angle has the smallest magnitude (is the most pointy).

Wireframe Graph

If a Polytope has vertices then its edges are either line-segments or half-lines (present only when the Polytope is not bounded). Hence with every bounded Polytope we can associate a graph consisting of its vertices and edges. We call it the Wireframe of the Polytope. A Wireframe can be projected to produce a plane drawing although this typically results in a distortion of the true length of the edges. To retain the true-length info in the drawing we can use the length of the original edge as the label of the projected edge. Similarly, we can define a Spherical Wireframe of a Solid Angle.

Angle Graph

The nodes of this graph represent Hyperfaces. 2 nodes are joined if the Hyperfaces are adjacent. (We consider 2 parallel Hyperplanes to be adjacent.) The edges are labeled with the dihedral angle between the Hyperfaces. Note that dihedral angles of a Polyhedron may also be used to label the edges of its Wireframe. A Wireframe with dihedral labels and a Coxeter graph are 2 different things. In the case of a Tetrahedron, one graph can be derived from the other by switching the labels on the opposite edges. For any bounded Polytope, its Dual is a Polytope which vertices are centers of the Hyperfaces of the original Polytope. The Wireframe of the Dual Polytope matches the (un-labeled) Angle graph of the orignal Polytope.

Simplex

A Simplex in an n-dimensional Hyperspace is a Polytope bound by n+1 Hyperplanes (which is the smallest number necessary for a Polytope to be bounded). It also has n+1 vertices. Both the Wireframe and the Angle graph of a Simplex has n+1 nodes where each pair is joined. A Simplicial Angle in an n-dimensional Hyperspace is a Solid Angle bound by n Hyperplanes. A solid angle at a vertex of a Simplex is a Simplicial Angle. Both the Spherical Wireframe and the Angle graph of the Simplicial Angle has n nodes where each pair is joined. Simplices and Spherical Shapes of their Simplicial Angles in low dimensions were given names

dimension	name
dimension	Simplex	Spherical Shape
1	Segment	Pole
2	Triangle	Arc
3	Tetrahedron	Spherical Triangle

Concept of Symmetry

Plane shapes can be manipulated: moved about or flipped. Such manipulations are called congruences. Concept of a congruence extends that of a primitive. Every primitive can be associated with a congruence. A line H can be associated with a congruence σ_H which flips (reflects) the plane using the line as a hinge. A point P can be associated with a congruence ρ_P which rotates the plane pinned down at the point by 180°. Such rotation is also called an inversion. The result of applying to 2 congruences in sucession is again a congruence, called their composite and denoted using multiplicative symbol ∘. In view of that, a (counterclockwise) rotation by 90° can be denoted by ρ_P^½. More generally, any rotation may denoted by ρ_P^θ for some θ between 0 and 2. There is no primitive which can be associated with the congruence of sliding (translating) the plane by a fixed amount in some direction. So, to be consistent, we invent a free-floating 'shape' called Vector. A model for a vector which points North would be a compass needle. We denote a translation corresponding to a vector v by τ^v. One could think that combining congruences of different types would lead to mixtures in arbitrary proportions. This turns out not to be the case. With one exception, any congruence can be described by a primitive, supplemented by either a vector or an angular parameter θ as described above. The exceptional mixture is a glide-reflection which is a composite of a reflection and a translation parallel to the line of reflection.

Symmetry Types

A symmetry of a plane drawing is any congruence which leaves the drawing unchanged. In the examples listed below, all symmetries are composite powers of a single congruence. The list gives a representative example for each congruence-type.

Translation
Inversion
Rotation
Reflection
Glide Reflection

The original drawings are in red. The black markers are not part of the drawings. They are primitives/vectors indicating presence of symmetry of a particular type. Drawings which symmetry indicators include a vector (1st and last) are assumed to repeat indefinitely to the left and to the right. In all cases there is nothing above or below the drawing, just an empty background. A choice to use 3-fold as a representative for any n-fold rotation is quite arbitrary, as is the lenght of the vectors.

Group of Symmetries

Any particular drawing may possess one symmetry, more than one, or none at all. One might think that by allowing drawings with any number of combined symmetries will lead to patterns of ever increasing complexity. Surprisingly, this is not the case. You can get an accurate picture of what symmetric patterns are possible just by listing 2 dozen examples or so. The reson behind it is two-fold

Discrete condition

Any composite relation satisfied by symmetries must be rational. If a drawing had 2 parallel vector symmetries v, w such that v : w were irrational then it would contain vector symmetries of arbitrary small length. Similarly, a symmetric rotation with an irrational angle would lead to symmetry rotations by arbitrary small angles.

Secondary Drawing

If we manage to find all indicators residing within a repeating portion of the drawing then the indicators for the remaining ones can be reconstructed simply be replication that portion and treating the indicatore as if they were part of the drawing. This is a result of the rule of composition which essentially translates into an operation of a congruence on a primitive/vector. An instance of this situation is when a drawing contains a point such that any symmetry of the drawing either fixes this point or is a composite of one such with a translation. If such point exists it is called Special.

Point Symmetries

When analyzing a symmetric pattern it is usually helpful to describe point-symmetries first. We simply create a figure by moving all cut-outs into one place without changing the tilt so that their centers coincide. This will produce a different drawing and a different (finite) set of symmetries. If a drawing has a special point then any point symmetry can be represented by an actual symmetry. The glide-reflection pattern does not have special points because it does not have fixed points at all.