Möbius Net


The perspective image of an infinite checkerboard. It can be constructed starting from any triangle DeltaOXY, where OX and OY form the near corner of the floor, and XY is the horizon (left figure). If OY_1P_(1,1)X_1 is the corner tile, the lines Y_1P_(1,1) and P_(1,1)X_1 must be parallel to OX and OY respectively. This means that in the drawing they will meet OX and OY at the horizon, i.e., at point X and point Y respectively (right figure). This property, of course, extends to the two bunches of perpendicular lines forming the grid.


The adjacent tile P_(1,1)X_1X_2P_(2,1) (left figure) can then be determined by the following conditions:

1. The new vertices X_2 and P_(2,1) lie on lines OX and Y_1P_(1,1) respectively.

2. The diagonal X_1P_(2,1) meets the parallel line OP_(1,1) at the horizon Z.

3. The line X_2P_(2,1) passes through Y.

Similarly, the corner-neighbor P_(11)P_(12)P_(22)P_(1,2) of OX_1P_(11)Y_1 (right figure) can be easily constructed requiring that:

1. Point P_(1,2) lie on X_1Y.

2. Point P_(2,2) lie on the common diagonal OP_(11) of the two tiles.

3. Line P_(2,2)P_(1,2) pass through X.


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


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 Y (left figure) and Z (right figure), so that


This entry contributed by Margherita Barile

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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.

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Möbius Net

Cite this as:

Barile, Margherita. "Möbius Net." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein.

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