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  2. Four-dimensional space - Wikipedia

    en.wikipedia.org/wiki/Four-dimensional_space

    Four-dimensional space (4D) is the mathematical extension of the concept of three-dimensional space (3D). Three-dimensional space is the simplest possible abstraction of the observation that one needs only three numbers, called dimensions, to describe the sizes or locations of objects in the everyday world.

  3. Manifold - Wikipedia

    en.wikipedia.org/wiki/Manifold

    In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an -dimensional manifold, or -manifold for short, is a topological space with the property that each point has a neighborhood that is homeomorphic to an open subset of -dimensional Euclidean space.

  4. Torus - Wikipedia

    en.wikipedia.org/wiki/Torus

    Poloidal direction (red arrow) and toroidal direction (blue arrow) A torus can be parametrized as: [2] (,) = (+ ⁡) ⁡ (,) = (+ ⁡) ⁡ (,) = ⁡. using angular coordinates , [,), representing rotation around the tube and rotation around the torus' axis of revolution, respectively, where the major radius is the distance from the center of the tube to the center of the torus and the minor ...

  5. Pythagorean theorem - Wikipedia

    en.wikipedia.org/wiki/Pythagorean_theorem

    In mathematics, the Pythagorean theorem or Pythagoras' theorem is a fundamental relation in Euclidean geometry between the three sides of a right triangle. It states that the area of the square whose side is the hypotenuse (the side opposite the right angle) is equal to the sum of the areas of the squares on the other two sides.

  6. Tetrahedron - Wikipedia

    en.wikipedia.org/wiki/Tetrahedron

    The tetrahedron is one kind of pyramid, which is a polyhedron with a flat polygon base and triangular faces connecting the base to a common point. In the case of a tetrahedron, the base is a triangle (any of the four faces can be considered the base), so a tetrahedron is also known as a "triangular pyramid".

  7. Rotations in 4-dimensional Euclidean space - Wikipedia

    en.wikipedia.org/wiki/Rotations_in_4-dimensional...

    In mathematics, the group of rotations about a fixed point in four-dimensional Euclidean space is denoted SO (4). The name comes from the fact that it is the special orthogonal group of order 4. In this article rotation means rotational displacement. For the sake of uniqueness, rotation angles are assumed to be in the segment [0, π] except ...

  8. 4-polytope - Wikipedia

    en.wikipedia.org/wiki/4-polytope

    A 4-polytope is a closed four-dimensional figure. It comprises vertices (corner points), edges, faces and cells. A cell is the three-dimensional analogue of a face, and is therefore a polyhedron. Each face must join exactly two cells, analogous to the way in which each edge of a polyhedron joins just two faces.

  9. Hypercube - Wikipedia

    en.wikipedia.org/wiki/Hypercube

    In geometry, a hypercube is an n -dimensional analogue of a square (n = 2) and a cube (n = 3). It is a closed, compact, convex figure whose 1- skeleton consists of groups of opposite parallel line segments aligned in each of the space's dimensions, perpendicular to each other and of the same length. A unit hypercube's longest diagonal in n ...

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