Geometry
... Shapes, both two and three-dimensional, exist in great variety. There are many different ways to see and describe similarities and differences among shapes. The more ways that one can classify and discriminate shapes, the better one understands them. Shapes have properties that can be used when ...
... Shapes, both two and three-dimensional, exist in great variety. There are many different ways to see and describe similarities and differences among shapes. The more ways that one can classify and discriminate shapes, the better one understands them. Shapes have properties that can be used when ...
Triangle reflection groups
... The main results of this paper are the introduction and partial proof of a theorem concerning the group of transformations generated by reflections in the sides of a triangle and a corollary concerning the finiteness of the group based on an algebraic property of one of its presentations. The main r ...
... The main results of this paper are the introduction and partial proof of a theorem concerning the group of transformations generated by reflections in the sides of a triangle and a corollary concerning the finiteness of the group based on an algebraic property of one of its presentations. The main r ...
Euclidean vs Non-Euclidean Geometry
... from each other even if extended to infinity, and are known as parallels. In hyperbolic geometry they "curve away" from each other, increasing in distance as one moves further from the points of intersection with the common perpendicular; these lines are often called ultraparallels. In ellipt ...
... from each other even if extended to infinity, and are known as parallels. In hyperbolic geometry they "curve away" from each other, increasing in distance as one moves further from the points of intersection with the common perpendicular; these lines are often called ultraparallels. In ellipt ...
Four-dimensional space
In mathematics, four-dimensional space (""4D"") is a geometric space with four dimensions. It typically is more specifically four-dimensional Euclidean space, generalizing the rules of three-dimensional Euclidean space. It has been studied by mathematicians and philosophers for over two centuries, both for its own interest and for the insights it offered into mathematics and related fields.Algebraically, it is generated by applying the rules of vectors and coordinate geometry to a space with four dimensions. In particular a vector with four elements (a 4-tuple) can be used to represent a position in four-dimensional space. The space is a Euclidean space, so has a metric and norm, and so all directions are treated as the same: the additional dimension is indistinguishable from the other three.In modern physics, space and time are unified in a four-dimensional Minkowski continuum called spacetime, whose metric treats the time dimension differently from the three spatial dimensions (see below for the definition of the Minkowski metric/pairing). Spacetime is not a Euclidean space.