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Penrose map

Mapping of the top left quadrant to the top left triangle of the Einsteins diamond. While the usual coordinate system on the left spans to infinity, the transform fits into finite triangle. Drag mouse to paint, this is possible both in quadrant and in the triangle. Press any key to clear.
Penrose map (or conformal compactifaction or Einstein's diamond) is a transform of usual Euclidean space into compact square. While Euclidean space is infinite, the result of this transform is finite and can be fully pictured on the screen or chart. Each quadrant of the Euclidean space is mapped into one quarter of the "diamond" (seen in the center):

Only top right quadrant and corresponding right triangle are shown in the applet.

The transform itself is described as[1]

This transform has the following features:

  • Lines with slope +-1 (called null rays in the literature about this transform) map into lines with the same slope (also to null rays).
  • The entire line at infinity gets maps to the diagonal border of the "diamond".
  • Horizontal and vertical lines on the diamond are inverse-mapped to hyperbolae on Minkowski space.

Unlike logarithmic scale (that is more commonly used to picture both very small and very big values in the same graph), Einstein diamond actually includes the infinity, bringing it to a finite distance. Also, unlike logarithmic scale, this transform has no problems with negative and zero value. At the same time, smaller local patterns (like pictured writing) are still possible to recognize after transform. However at the time of writing it seems more common to use this mapping for fundamental analysis than for visualizations.

This transform is useful in dealing with the global behaviour of non-linear wave equations[2]. When using it this way, the vertical (y) coordinate is the time and horizontal (x) coordinate is the (single dimensional) space. Penrose map can be extended to higher dimensions.

References

  1. 1 Conformal Compactifaction applet by Terence Tao
  2. 2 Global well-posedness for large data of one-dimensional wave maps below the energy norm", with Mark Keel, submitted to IMRN. dvi* Figure 1 ps.