it; the line segment is the 1-simplex. The line segment itself was formed by starting with a single point in 0-dimensional space (this initial point is the 0-simplex) and adding a second point, which required the increase to 1-dimensional space. More formally, an (n + 1)-simplex can be constructed as a join (∨ operator) of an n-simplex and a point, ( ). An (m + n + 1)-simplex can be constructed as a join of an m-simplex and an n-simplex. The two simplices are oriented to be completely normal from each other, with translation in a direction orthogonal to both of them. A 1-simplex is the join of two points: ( ) ∨ ( ) = 2 ⋅ ( ). A general 2-simplex (scalene triangle) is the join of three points: ( ) ∨ ( ) ∨ ( ). An isosceles triangle is the join of a 1-simplex and a point: { } ∨ ( ). An equilateral triangle is 3 ⋅ ( ) or {3}. A general 3-simplex is the join of 4 points: ( ) ∨ ( ) ∨ ( ) ∨ ( ). A 3-simplex with mirror symmetry can be expressed as the join of an edge and two points: { } ∨ ( ) ∨ ( ). A 3-simplex with triangular symmetry can be expressed as the join of an equilateral triangle and 1 point: 3.( )∨( ) or {3}∨( ). A regular tetrahedron is 4 ⋅ ( ) or {3,3} and so on. In some conventions, the empty set is defined to be a (−1)-simplex. The definition of the simplex above still makes sense if n = −1. This convention is more common in applications to algebraic topology (such as simplicial homology) than to the study of polytopes. Symmetric graphs of regular simplices These Petrie polygons (skew orthogonal projections) show all the vertices of the regular simplex on a circle, and all vertex pairs connected by edges. The standard simplex The standard n-simplex (or unit n-simplex) is the subset of Rn+1 given by The simplex Δn lies in the affine hyperplane obtained by removing the restriction ti ≥ 0 in the above definition. The n + 1 vertices of the standard n-simplex are the points ei ∈ Rn+1, where e0 = (1, 0, 0, ..., 0), e1 = (0, 1, 0, ..., 0), ⋮ en = (0, 0, 0, ..., 1). A standard simplex is