# Taneli Luotoniemi

Doctoral candidate of art

Dept. of Mathematics and Systems Analysis, Aalto University

Helsinki

Stericated 5-simplex is the expanded form of the 5-dimensional simplex. When it is projected from its center to a four-dimensional picture space, the result is a configuration of fifteen points, twenty lines, fifteen planes and six 3-spaces.

The pair of models both depict the same configuration, but are painted differently to embody projective theorems about triangles and tetrahedra. They are generalizations of the reowned Desargues theorem, which states that if two triangles are in perspective with respect to a point (the center of perspectivity) i.e. the lines connecting their vertices are concurrent, then they are also in perspective with respect to a line, i.e. the intersections of the extensions of corresponding edges are collinear.

The pair of models both depict the same configuration, but are painted differently to embody projective theorems about triangles and tetrahedra. They are generalizations of the reowned Desargues theorem, which states that if two triangles are in perspective with respect to a point (the center of perspectivity) i.e. the lines connecting their vertices are concurrent, then they are also in perspective with respect to a line, i.e. the intersections of the extensions of corresponding edges are collinear.

Perspectivity Between Two Tetrahedra

50 x 50 x 50 cm

Painted wooden rods

2018

If two tetrahedra are in perspective from a point (white), then the intersections of the extensions of the corresponding edges are coplanar (black).

Perspectivity Between Three Triangles

50 x 50 x 50 cm

Painted wooden rods

2018

If the three centers of perspectivity (orange, green, purple) determined by three triangles (blue, pink, yellow) are collinear (black), then the intersections of the extensions of corresponding edges are collinear (white) as well.