{{Technical, date=February 2024 A tetraview is an attempt to

graph Graph may refer to: Mathematics *Graph (discrete mathematics), a structure made of vertices and edges **Graph theory, the study of such graphs and their properties *Graph (topology), a topological space resembling a graph in the sense of discret ...

complex function Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is helpful in many branches of mathematics, including algebraic g ...

of a complex variable, by a method invented by Davide P. Cervone. A graph of a

real function In mathematical analysis, and applications in geometry, applied mathematics, engineering, and natural sciences, a function of a real variable is a function whose domain is the real numbers \mathbb, or a subset of \mathbb that contains an inter ...

of a real variable is the

set Set, The Set, SET or SETS may refer to: Science, technology, and mathematics Mathematics *Set (mathematics), a collection of elements *Category of sets, the category whose objects and morphisms are sets and total functions, respectively Electro ...

ordered pair In mathematics, an ordered pair, denoted (''a'', ''b''), is a pair of objects in which their order is significant. The ordered pair (''a'', ''b'') is different from the ordered pair (''b'', ''a''), unless ''a'' = ''b''. In contrast, the '' unord ...

s (x,y) such that y = f(x). This is the ordinary two-

dimension In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coo ...

al Cartesian graph studied in school

algebra Algebra is a branch of mathematics that deals with abstract systems, known as algebraic structures, and the manipulation of expressions within those systems. It is a generalization of arithmetic that introduces variables and algebraic ope ...

. Every

complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the for ...

has both a

real part In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form ...

and an

imaginary part In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form ...

, so one complex variable is two-dimensional and a BBC pair of complex variables is four-dimensional. A tetraview is an attempt to give a picture of a four-dimensional object using a two-dimensional representation—either on a piece of paper or on a computer screen, showing a still picture consisting of five views, one in the center and one at each corner. This is roughly analogous to a picture of a three-dimensional object by giving a front view, a side view, and a view from above. A picture of a three-dimensional object is a

projection Projection or projections may refer to: Physics * Projection (physics), the action/process of light, heat, or sound reflecting from a surface to another in a different direction * The display of images by a projector Optics, graphics, and carto ...

of that object from three dimensions into two dimensions. A tetraview is set of five projections, first from four dimensions into three dimensions, and then from three dimensions into two dimensions. A complex function w = f(z), where z = a + b''i'' and w = c + d''i'' are complex numbers, has a graph in four-space (four dimensional space) R⁴ consisting of all points (a, b, c, d) such that c + d''i'' = f(a + b''i''). To construct a tetraview, we begin with the four points (1,0,0,0), (0, 1, 0, 0), (0, 0, 1, 0), and (0, 0, 0, 1), which are vertices of a spherical

tetrahedron In geometry, a tetrahedron (: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular Face (geometry), faces, six straight Edge (geometry), edges, and four vertex (geometry), vertices. The tet ...

on the unit three-sphere S³ in R⁴. We project the four-dimensional graph onto the three-dimensional sphere along one of the four

coordinate axes In geometry, a coordinate system is a system that uses one or more numbers, or coordinates, to uniquely determine and standardize the position of the points or other geometric elements on a manifold such as Euclidean space. The coordinates are ...

, and then give a two-dimensional picture of the resulting three-dimensional graph. This provides the four corner graph. The graph in the center is a similar picture "taken" from the point of view of the origin.

External links

* http://www.math.union.edu/~dpvc/professional/art/tetra-exp.html Functions and mappings