Higher-dimensional geometry

The geometry of spaces of dimension more than three; the term is applied to those spaces whose geometry was initially developed for the case of three dimensions and only later was generalized to a dimension $ n > 3 $; first of all the Euclidean spaces and then the Lobachevskii, Riemannian, projective, affine, and pseudo-Euclidean spaces. (The general Riemannian and other spaces were defined at once for $ n $ dimensions. See also Affine space; Euclidean space; Lobachevskii space; Projective space; Pseudo-Euclidean space; Riemannian space.) At present the separation of three-dimensional and higher-dimensional geometry has mainly historical and pedagogical significance, since problems can be posed and solved for any number of dimensions, when, and so long as, they are meaningful. The construction of the geometry of the spaces mentioned for $ n $ dimensions is carried out in a similar way to the three-dimensional case. In this connection it is possible to proceed directly from a generalization of the geometric foundation of three-dimensional geometry, from certain axiom systems or from a generalization of analytic geometry, translating its basic results from the case of three coordinates to arbitrary $ n $. This is exactly how the construction of $ n $-dimensional Euclidean geometry was begun.

The historical representation of spaces of more than three dimensions came gradually, primarily on the grounds of the geometric representation of powers: $ a ^ {2} $ is the "square" , $ a ^ {3} $ is the "cube" , but $ a ^ {4} $, etc. no longer have a graphic representation and it was said that $ a ^ {4} $ is "biquadratic" , $ a ^ {5} $ is "cubo-quadratic" , etc. (as long ago as Diophantus in the 3rd century, and later by a number of medieval authors). The idea of a higher-dimensional space was expressed by I. Kant (1746), while J. d'Alembert (1764) wrote on attaching to space the time as a fourth coordinate. The construction of $ n $-dimensional geometry was accomplished by A. Cayley (1843), H. Grassmann (1844) and L. Schläfli (1852). The initial doubts and mysticism associated with the merging of these generalizations with physical space were overcome, and $ n $-dimensional space as a fruitful formal mathematical idea has been completely consolidated into mathematics.

Euclidean space of arbitrary dimension $ n \geq 3 $ (not excluding the infinite-dimensional case) is easiest of all defined as that in which there are distinguished subsets, namely lines and planes, with the usual relations: membership, order, congruence (either defined by distance or by motion) and in which all the usual axioms are satisfied, except the following: Two planes having a common point have at least one more common point. If this is satisfied, the space must be three-dimensional; if it is not satisfied, so that there are two planes with a unique common point, then the space is at least four-dimensional.

The notion of a plane is generalized in the following way: A flat is a set of points which together with any two of its points contains the line passing through them. In this sense all spaces are also flats. The intersection of all flats containing a given set $ M $ is the flat "spanned by M" (the affine hull of $ M $). If a flat is spanned by $ m + 1 $ points but not by any smaller number of them, then it is called $ m $-dimensional or, briefly, an $ m $-flat. A point is a $ 0 $-flat, a line is a $ 1 $-flat, an ordinary plane is a $ 2 $-flat, three-dimensional space is a $ 3 $-flat. A space is called $ n $-dimensional if it is an $ n $-flat. That is, for the definition of the $ n $-dimensional Euclidean space $ E _ {n} $, for any given $ n \geq 3 $, it is sufficient to add the axiom: The space is an $ n $-flat. In it there is an $ m $-flat for each $ 0 \leq m \leq n - 1 $. Each $ m $-flat with $ m \geq 2 $ is an $ m $-dimensional Euclidean space $ E _ {m} $. Since 4 points are always contained in a $ 3 $-flat, any two lines are contained in a $ 3 $-flat, that is, in $ E _ {3} $.

In $ E _ {n} $ through any point it is possible to draw $ n $, and no more, mutually perpendicular lines and to introduce corresponding rectangular coordinates $ x _ {1} \dots x _ {n} $; in these the length of a segment $ X Y $ is expressed by the formula

$$ \tag{* } X Y = \sqrt { ( x _ {1} - y _ {1} ) ^ {2} + \dots + ( x _ {n} - y _ {n} ) ^ {2} } . $$

Formula (*) can be used as the basis for a coordinate definition of $ E _ {n} $, equivalent to the previous definition. Namely: $ E _ {n} $ is a set in which coordinates $ x _ {1} \dots x _ {n} $ (taking all possible values) have been introduced and with each pair of points $ X ( x _ {1} \dots x _ {n} ) $ and $ Y ( y _ {1} \dots y _ {n} ) $ there is associated a "distance" , which is the number (*); here, as the geometry of $ E _ {n} $ are taken those and only those definitions and statements that can be formulated in terms of the distance relation. For example, a segment $ A B $ is the set of all points $ X $ for which $ A X + XB = A B $, and the line $ A B $ is the set of all $ X $ for which $ \pm A X \pm X B = A B $.

Vector calculus is constructed in $ E _ {n} $ as in $ E _ {3} $ (starting from the geometrical or the coordinate definition); the difference is simply that in $ E _ {n} $ a vector has $ n $ components (correspondingly, $ n $ vectors may be independent). For example, the inner product is:

$$ ( \mathbf a , \mathbf b ) = \ | \mathbf a | \cdot | \mathbf b | \cos \alpha = \ \sum a _ {i} b _ {i} . $$

But the vector product cannot be defined for $ n > 3 $, since a $ 2 $-flat has perpendiculars in different directions (those passing through a point fill out an $ ( n - 2 ) $-flat). Instead of vector product the notion of a bivector is used. A combination of direct geometric, coordinate and vector methods gives the most complete arsenal of means for the development of the geometry of $ E _ {n} $. The geometric approach allows one to transfer immediately to $ E _ {n} $ planimetry and stereometry, that is, the geometry of 2- and $ 3 $-flats, and further to construct the stereometry of $ E _ {n} $ itself, which naturally generalizes the stereometry of $ E _ {3} $: theorems on perpendiculars, parallel planes, etc. For example, a line perpendicular to $ m $ lines in an $ m $-flat is perpendicular to every line in that flat. Many definitions and proofs are given for $ E _ {n} $ by induction on $ n $. For example, an $ n $-dimensional polytope, or $ n $-polytope, is a body (a closed bounded domain in $ E _ {n} $) whose boundary consists of a finite number of $ ( n - 1 ) $-polytopes (cf. also Polygon; Polyhedron; Regular polygons; Regular polyhedra). The simplest polytopes are: a prism, filled out by equal parallel segments drawn from all the points of an $ ( n - 1) $-polytope, a pyramid, filled out by segments drawn from one point to all points of an $ ( n - 1 ) $-polytope; the simplest of these are: the $ n $-cube; a right prism, whose faces are $ ( n - 1 ) $-cubes (a $ 2 $-cube is a square); and the $ n $-simplex with an $ ( n - 1 ) $-simplex as its base (a $ 2 $-simplex is a triangle). Content is defined in the same way as volume in $ E _ {3} $. Thus, in $ E _ {n} $, there are $ n $ contents: $ 1 $-content is length, $ 2 $-content is area, etc. For a prism the $ n $-content is $ V = S h $ and for a pyramid $ V = S h / n $, where $ S $ is the $ ( n - 1 ) $-content of the base and $ h $ is the height. A vast and well-developed domain of the geometry of $ E _ {n} $ is the theory of convex bodies.

It is possible to distinguish three kinds of facts about higher-dimensional geometry: 1) those that are direct generalizations from $ E _ {3} $ (for example, the theorem on content just mentioned); 2) those that correspond to analogous facts for various dimensions $ m \leq n $ (for example, a convex body with a centre of symmetry is uniquely determined by the $ m $-contents of its $ m $-dimensional projections, for any $ m \geq 1 $ and $ m < n $); and 3) those that display the essential difference between different $ E _ {n} $ (for example, the number of regular polytopes in $ E _ {3} $ is equal to 5, in $ E _ {4} $ it is equal to 6, and in $ E _ {n} $, for $ n \geq 5 $, there are three of them: the simplex, the cube and the cross polytope, which is the analogue of the octahedron; another example: a convex polyhedral (not trihedral) angle in $ E _ {3} $ is flexible, in $ E _ {n} $ for $ n > 3 $ it is always rigid; the theories of surfaces in $ E _ {3} $ and in $ E _ {n} $ for $ n > 3 $ are essentially different).

The Lobachevskii spaces $ \Lambda _ {n} $ and the affine spaces $ A _ {n} $ are defined completely analogously to $ E _ {n} $. $ \Lambda _ {n} $ satisfies the same axioms as $ E _ {n} $, with the parallel axiom changed as in $ \Lambda _ {2} $, and in $ A _ {n} $ all the axioms of $ E _ {n} $ are satisfied except the axioms of congruence, and the notion of congruence itself is excluded. Analogously, by variation of the axioms of incidence it is possible to define the $ n $-dimensional projective space $ P _ {n} $. Another way of defining such a space is to introduce coordinates and to give its group of transformations; then geometric relations are those and only those that are invariant relative to this group. In the case of $ E _ {n} $ this is the group of similarities (compositions of orthogonal transformations and dilatations); for $ A _ {n} $ it is the group of all linear (inhomogeneous) transformations; see also Projective geometry; Lobachevskii geometry.

Pseudo-Euclidean spaces can be defined by coordinates: $ E _ {n} ^ {n- m} $ is a set with coordinates $ x _ {1} \dots x _ {n} $ and the "interval" between two points $ X $ and $ Y $ is the square root of

$$ ( x _ {1} - y _ {1} ) ^ {2} + \dots + ( x _ {m} - y _ {m} ) ^ {2} + $$

$$ - ( x _ {m- 1} - y _ {m- 1} ) ^ {2} - \dots - ( x _ {n} - y _ {n} ) ^ {2} ; $$

those definitions and statements are to be regarded as geometric that can be formulated in terms of intervals; in other words, those that are invariant relative to the group of transformations preserving the relationships of intervals. In the special theory of relativity space-time (i.e. Minkowski space) is defined as $ E _ {4} ^ {1} $; in it the interval is

$$ c ^ {2} ( t _ {1} - t _ {2} ) ^ {2} - ( x _ {1} - x _ {2} ) ^ {2} - ( y _ {1} - y _ {2} ) ^ {2} - ( z _ {1} - z _ {2} ) ^ {2} , $$

where $ t $ is the time, $ x , y , z $ are spatial coordinates in the given frame of reference and $ c = \textrm{ const } $ is the velocity of light.

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In other words, a flat is an affine subspace (cf. also Affine space).

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