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Opposite angles of a rhombus have equal measure. The two diagonals of a rhombus are perpendicular; that is, a rhombus is an orthodiagonal quadrilateral. Its diagonals bisect opposite angles. The first property implies that every rhombus is a parallelogram.
The concept of antipodal points is generalized to spheres of any dimension: two points on the sphere are antipodal if they are opposite through the centre.Each line through the centre intersects the sphere in two points, one for each ray emanating from the centre, and these two points are antipodal.
For example, the dimension of a point is zero; the dimension of a line is one, as a point can move on a line in only one direction (or its opposite); the dimension of a plane is two etc. The dimension is an intrinsic property of an object, in the sense that it is independent of the dimension of the space in which the object is or can be embedded.
Kites and isosceles trapezoids are dual to each other, meaning that there is a correspondence between them that reverses the dimension of their parts, taking vertices to sides and sides to vertices. From any kite, the inscribed circle is tangent to its four sides at the four vertices of an isosceles trapezoid.
A stereographic projection of a Clifford torus in four dimensions performing a simple rotation through the xz-plane. The torus has a generalization to higher dimensions, the n-dimensional torus, often called the n-torus or hypertorus for short. (This is the more typical meaning of the term "n-torus", the other referring to n holes or of genus n ...
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The opposite or facing sides of a parallelogram are of equal length and the opposite angles of a parallelogram are of equal measure. The congruence of opposite sides and opposite angles is a direct consequence of the Euclidean parallel postulate and neither condition can be proven without appealing to the Euclidean parallel postulate or one of ...
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