Try rotating things.

Aaron Sloman

http://www.cs.bham.ac.uk/~axs

School of Computer Science University of Birmingham,UK

This paper is

http://www.cs.bham.ac.uk/research/projects/cogaff/misc/discrete-rotate.html

(A PDF version may be added later.)

A partial index of discussion notes is in

http://www.cs.bham.ac.uk/research/projects/cogaff/misc/AREADME.html
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There are, moreover, physicists who challenge that orthodox view. An example is

David Tong, The Unquantum Quantum,Scientific American, 307/6, December, 2012, pp. 46--49,

Note: The examples below were produced on a version of 32-bit linux, using 32-bit graphical software. Things have changed since then: the same tests run on 64 bit linux do not produce these visible distortions, owing to the enormous increase in precision.

How does everyday experience challenge the orthodox view?

Anyone who has tried to implement algorithms for rotating digitised images stored in computers will have discovered that it is impossible to have structure-preserving combinations of rotations about arbitrarily chosen locations except in very simple cases, such as rotations that are multiples of 90 degrees.

E.g. if you have a binary image represented in a 2-D array and you perform rotations of various amounts about various points in the image you will not in general be able to reverse those rotations and get back the original image. Even rotation in one direction by 7 degrees several times around the same point, then performing the same number of rotations by -7 degrees will not normally restore the image. (Any number that is not a multiple of 90 will suffice.)

What happens will depend on whether the image is represented in a binary array, or represented in terms of coordinates and equations for lines, arcs, etc. The rotation of the binary array will lose structure when a grid location that should be partly black and partly white after a rotation is forced to be all black or all white, The rotation of the image stored algebraically will lose structure because coordinates will be truncated after rotation because of the limited size of the computer. For example, if a rotation changes the X coordinate of a point to the square root of 2, then that value cannot be represented by a finite number of bits (using standard methods for representing real numbers). An example would be rotating a 1 by 1 square clockwise by 45 degrees about its bottom left corner, based on the coordinate origin.

To illustrate effects of rotations actually performed on a computer, here's an image (FIG 1) drawn using the TGIF drawing package, then saved as a digital image:

**FIG 1**

FIG 2 below is the result of using TGIF to rotate counter clockwise several times by 7 degrees, then clockwise by the same number of degrees. In this case the intermediate stages are represented by drawing commands with numerical arguments, rather than by binary arrays.

**FIG 2**

In the figure above, corners that previously coincided in the image, and corners and curves that previously coincided, have now moved apart, with corresponding changes along the lengths of the lines involved.

This is due to the fact that floating point numbers (point coordinates) were represented using only 32 bit arithmetic, so that the rounding errors were quite large. Using 64 or 128 bit arithmetic would reduce the rounding errors, but not remove them.

FIG 3, below, is the image after clipping the picture to leave the image asymmetrically located in its frame, then saving as a digitised image.

**FIG 3**

FIG 4, below, is the result of rotating that clipped image several times about its new centre, rotating counter-clockwise by 7 degrees then clockwise by the same number of degrees, using the 'xv' package. I.e. the rotation is done using the digitised image, not equations and coordinates. In this case the result of the rotations is blurring of the image. The black surround is a result of copying 'unknown' image contents from outside the frame during the rotation.

**FIG 4**

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**Related discussion note, written a year later:**

http://www.cs.bham.ac.uk/research/projects/cogaff/misc/quantum-non-quantum-rotation.html

"Letter to a quantum theorist"

(When I wrote that, I had forgotten about this earlier document! I later added
a link to this paper.)

Also related:

http://www.cs.bham.ac.uk/research/projects/cogaff/misc/construction-kits.html

Construction kits required for biological evolution

(Including evolution of minds and mathematical abilities.)

The scientific/metaphysical explanatory role of construction kits
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**Maintained by**

Aaron Sloman

http://www.cs.bham.ac.uk/~axs