Post Script to An Odd Visual Effect in a Tibetan Rug

After, we had published and announced Part 2 in this sequence of posts, I received two additional submissions.  I am not going to announce it, but am simply publishing this postscript so as to include them.

First, Pat Weiler wrote saying  “I am pretty sure that this optical illusion has been present in other art forms prior to this use, such as tile work and mosaics.”

He sent along a “modern” but “related” further example.

stair rug

And Dave Scherbel wrote:

“…You guys are working very hard at this.  Let me propose a very simple explanation.  Please refer to the photoshopped figure below of your “head on” shot.

annotated tibetanrugstraighton copy

“There are twice as many blue squares than white ones. Forget the light/dark blue issue: when seen from a foreshortened view, they appear as just blue. The geometry is such that when you stand away from the center of the pattern but along a 45 degree view (such as the “perspective” view photograph), you see blue parallel lines extending along your line of sight. That is because the pattern creates a continuous unbroken line with a width 1/2 the diagonal of the squares.  See the parallel lines I’ve drawn on the rug with the red arrows indicating line of sight. This effect happens when viewing the upper right and lower left quadrants. 

“However, when viewing the upper left and lower right quadrants, the continuous blue lines extend basically right to left to line of sight.  See black arrow showing line of sight extending perpendicular to the continuous blue line in the upper left.  When you return to the “heads on” view, you are deprived of the any sight line of a continuous blue line with the width 1/2 the diagonal of the squares.

“Understand that foreshortening has a role to play in this also.  From the “perspective view”, each continuous blue line, parallel to your line of sight, is visually squeezed, top to bottom, making the blue lines darker and a stronger identity.  In contrast, when viewing the top and bottom quadrants, the foreshortening only effect each individual square separately, thus maintaining the spacial identity of the white squares relative to the blue line. In the parallel line view, the white squares only seem to give the continuous blue lines a saw-tooth edge look.

“Now here’s the fun part.  This only works because the rug viewed is relatively small.  Consider a very large rug, large room size, with the same sized squares.  Stand on the same 45 degree line  from the center, but further away from the center.  Now, the view will change dramatically as you look from the center to the right or left.  Looking parallel to the right side of the rug you will now be looking at the squares pattern effectively in the “heads on” view and will not experience the blue lines: in front of you you will be seeing that part of the rug as experience in your “head on” view.  Moving your view back to the left towards the center, at some point in the rotation, the lines will seem to re-appear again and you will see the center section of the rug in the “perspective” view.”

I thanked David for this, and then wrote him:

“It seems like a more detailed and extensive version of Matthew Polk’s explanation based on his “lines” replication.  But it’s clearly a useful, “visually-cued” elaboration and one based on the rug itself.

“If we treat the two blues as one, does that convert the image to a 2×2 format, and, if so, why doesn’t the effect seem to be visible in William Bateson’s 2×2 replication?”

David said:

“In response to your question, I took William’s three color diagram (white/gray/black) and converted the gray squares to black, so it’s only two “color” now.

gray to black

“As you can see …, except at a few selected 2by2 boxes in the pattern (e.g., near the center), it does not convert to 2×2: it’s something like 33×67 in a 10×10 box; 8×17 in a 5by5 box; 5×11, in a 4by4 box; and 3×1 in a 2by2 box. 

“So the ratio varies depending on the number of squares on the sides of the selected box,  but most likely is never same/same above 2×2 boxes near where the pattern changes directions.

“However, I believe it diminishes any solutions involving the blue vs light-blue squares.”

My thanks to Pat and David for these additional contributions.

Regards,

R. John Howe

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