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The space groups in two dimensions are called wallpaper groups. In three dimensions, the space groups are the symmetry groups possible in a crystal lattice with the ...

If the cross ratio kappa of {AB,CD} satisfy kappa^2-kappa+1=0, (1) then the points are said to form a bivalent range, and {AB,CD}={AC,DB}={AD,BC}=kappa (2) ...

A convex set K is centro-symmetric, sometimes also called centrally symmetric, if it has a center p that bisects every chord of K through p.

A chord which is a normal at each end. A centrosymmetric set K subset R^d has d double normals through the center (Croft et al. 1991). For a curve of constant width, all ...

Any two ranges {ABC...} and {A^'B^'C^'...} which are situated on the same or different lines are said to be homographic when the cross ratio of any four points on one range ...

A linear transformation of period two. Since a linear transformation has the form, lambda^'=(alphalambda+beta)/(gammalambda+delta), (1) applying the transformation a second ...

The isogonal transform of a geometric object is the object obtained by collectively taking the isogonal conjugates of all its points.

The isotomic transform of a geometric object is the object obtained by collectively taking the isotomic conjugates of all its points.

Any cubic curve that passes through eight of the nine intersections of two given cubic curves automatically passes through the ninth.

A geometry in which Archimedes' axiom does not hold.