Symmetries Of Checkered Cross SectionsWritten By Adi Cox 25th June 2015

IndroductionThis paper is analogous to looking at the cross sections of a cone to find second degree polynomial curves in the form of parabolas, hyperbolas and ellipses.The cone is replaced by the checkered cube and the various second degree polynomial curves are replaced by cross sections that have some of the seventeen types of wallpaper symmetries. Just how many is what this paper intends to find out.

Below is a checkered cube 10 x 10. The easiest and most obvious cross section would be translation z = 0, angle x = 0, angle y = 0, and so we would get the checkered squares which would be the wallpaper symmetry p4mm.

If I was to cut through the cube from the base corner at the front at angle pi/4 radians (45 degrees) to the horizontal. So that translation z = 0, angle x = pi/4, angle y =pi/4, then we would get this cross section :

If we look at the wallpaper classification algorithm below then we see that the cross section has the same symmetry as p3m1

Wallpaper Classification Algorithm

There are seventeen types of wallpaper patterns. To use the algorithm first determine the highest order of rotational symmetry and then follow the appropriate diagram below.

Standard forms and notation for wallpaper groups.nNotationPoint group1p1C12pmD13pgD14cmD15p2C16p2mm or pmm D17p2mgD28p2gg or pggD29c2mm or cmmD210p3C311p3m1D312p31mD313p4C414p4mm or p4mD415p4gm or p4gD416p6C617p6mm or p6mD6

All the cross sections of the checkered cube will be flat planes which will be determined by three factors only:

(1) The angle x that is horizontal to the x axis and arcs up to the vertical z axis.(2) The angle y that is horizontal to the y axis and arcs up to the vertical z axis.(3) Translation z that is a fraction of the check cube side length, and z is vertical along the z axis.

If I was to cut through the cube from half way up the base corner cube at the front at an angle of pi/4 radians (45 degrees) to the horizontal then we would get this cross section above. So translation z = 1/2, angle x = pi/4, angle y = pi/4.If we look at the wallpaper classification algorithm then we see that the cross section has the same symmetry as p6mm.