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Volume 9, Issue 2

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Spatial Inversion: Reflective Anamorphograms
Philip W. Kuchel

Inverting the Union Jack

The British flag (or Union Jack) is especially striking because of its numerous angles, bright colors, and yet geometrically simple construction. First, the flag is defined geometrically by representing it as a set of triangles, parallelograms, and more general polygons, in which the sides are further divided into short linear segments. After the points are transformed, the sides of the resulting polygonal sections will be good approximations to smooth curves. Figure 3 shows such a representation of the Union Jack.

The Union Jack is not completely symmetric: it is intended to have a well-recognized top and bottom edge. In times of military or civilian emergency, it can be flown upside down to attract attention. Nevertheless it has two-fold rotational symmetry about its center so only two Cartesian quadrants need be represented as a list of coordinates, and the other two can be obtained by the appropriate rotation. Specifically, the upper-right quadrant is rotated through to give the lower-left one, and the lower-right quadrant is rotated through to give the upper-left one. The transformation that delivers this rotation through the angle Alpha centered about the origin is

and of course in the present case .

The actual function used in the Mathematica program is called rotator and is implemented as follows.

Figure 3. The Union Jack.

The inverted flag is produced by subjecting the List of coordinates to the transformations expressed in equations (2) to (5).

The inverted flag can be optically transformed back to the undistorted image of the Union Jack by placing a reflecting cone of the right height and radius at its center and viewing the whole setup from above the apex of the cone. Thus, we will have inverted the Union Jack mathematically and reinverted it optically (Figure 4).

Figure 4. The reflective inversion of the Union Jack obtained by applying the coordinate transformation given by equations (2) to (5) to the polygonal coordinates that define Figure 3. This figure is inverted back to an undistorted Union Jack by a straight-sided cone that has a base circle that touches the inside of the central four cusps of the pattern and is of a height that is 8/7 of the diameter of the base; that is, the solid angle of the cone is .



     
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