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Möbius Net

The perspective image of an infinite checkerboard. It can be constructed starting from any triangle , where and form the near corner of the floor, and is the horizon (left figure). If is the corner tile, the lines and must be parallel to and respectively. This means that in the drawing they will meet and at the horizon, i.e., at point and point respectively (right figure). This property, of course, extends to the two bunches of perpendicular lines forming the grid.

The adjacent tile (left figure) can then be determined by the following conditions:

1. The new vertices and lie on lines and respectively.

2. The diagonal meets the parallel line at the horizon .

3. The line passes through .

Similarly, the corner-neighbor of (right figure) can be easily constructed requiring that:

1. Point lie on .

2. Point lie on the common diagonal of the two tiles.

3. Line pass through .

Iterating the above procedures will yield the complete picture. This construction shows how naturally projective geometry arises from perspective design, since and can be interpreted as two coordinate axes in the real projective plane with and their points at infinity, joined by the line at infinity .

The Möbius net is the result of a projective transformation of the two-dimensional lattice. Unlike in affine geometry, the length proportions along the two perpendicular directions are not preserved, whereas the cross ratio, which is invariant by central projection, is. The "horizontal" sides of the projected tiles have different lengths, but are related by the central projections from (left figure) and (right figure), so that

 (1) (2)

This entry contributed by Margherita Barile

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References

Fauvel, J.; Flood, R.; and Wilson, R. J. (Eds.). Möbius and his Band: Mathematics and Astronomy in Nineteenth-Century Germany. Oxford, England: Oxford University Press, p. 91, 1993.

Möbius Net

Cite this as:

Barile, Margherita. "Möbius Net." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/MoebiusNet.html