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 discr ...

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 ...

of a

complex variable 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 ...

, 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 of

ordered pair In mathematics, an ordered pair (''a'', ''b'') is a pair of objects. The order in which the objects appear in the pair is significant: the ordered pair (''a'', ''b'') is different from the ordered pair (''b'', ''a'') unless ''a'' = ''b''. (In co ...

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 coordin ...

al Cartesian graph studied in school

algebra Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics. Elementary a ...

. Every complex number 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 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 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 (plural: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners. The tetrahedron is the simplest of all the ...

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 A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in ...

, and then give a two-dimensional picture of the resulting

three-dimensional graph A three-dimensional graph may refer to * A graph (discrete mathematics), embedded into a three-dimensional space * The graph of a function of two variables In mathematics, the graph of a function f is the set of ordered pairs (x, y), where f(x) ...

. 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 * http://www.maa.org/cvm/1998/01/sbtd/article/tour/tetra-Z3/tetra-Z3.html Functions and mappings