{"id":8036,"date":"2023-11-21T15:52:04","date_gmt":"2023-11-21T15:52:04","guid":{"rendered":"https:\/\/handandmachine.org\/classes\/computational_fabrication\/?p=8036"},"modified":"2023-11-21T15:52:04","modified_gmt":"2023-11-21T15:52:04","slug":"eschers-escher-lauren","status":"publish","type":"post","link":"https:\/\/handandmachine.org\/classes\/computational_fabrication\/2023\/11\/21\/eschers-escher-lauren\/","title":{"rendered":"Escher’s Escher – Lauren"},"content":{"rendered":"\n
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2D Tiling<\/h2>\n\n\n\n

For the Escher Tiles, I was very inspired by Escher’s piece below, Angels and Devils<\/em>. <\/p>\n\n\n\n

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The original tiling is in hyperbolic space, but I based mine off of this square 2D Euclidean version to the right – I drew a square to emphasize the base tile.<\/p>\n\n\n\n

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I created my two lines in the shapes of where the heads of the demons and angels intersect, as shown in the first image in green. I noticed instead of the parallel line in the square being the same, like the tiling we did in class, it was the perpendicular line. This yields tiles that rotate 90 degrees every time. I found I could represent this by just mirroring the parallel lines for every lattice unit where (i+j)%2==1, or in a checker-boarded pattern. You can see this in the middle picture above, where the pink lines are versions of the green lines that have been flipped across the orange axes. I used rs.RotateObject(…) for this. When the lattice is constructed, you see the tiles that flip-flop direction in a checker board pattern – as in the third picture. Of course, this yields not quite the same pattern. I only created one tile that encompasses two devils and two angels, whereas Escher had two different tiles – one for the pair of angels and one for the pair of devils. <\/p>\n\n\n\n

Below is the individual tile, a 4×4 matrix, and a visualization of the shapes of the devils and angels within a tile.<\/p>\n\n\n\n

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For the second part, I chose to do triangles. Essentially, I created a triangle from 3 lines, with two vectors, as seen below. Then, in the horizontal direction, I moved every triangle half <\/em>of the vector length, and for every other triangle, (if j%2==0), I flipped it upside down. In the vertical direction, the triangle would move the full length.<\/p>\n\n\n\n

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Printed 3D Tiles<\/h2>\n\n\n
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I realized that if I formed the vertical curve as the intersection of the bottom of the devil \/ the side of the angel, that I would produce an angel tile that formed the devil in the negative space. I chose to loft my original Escher tile to this angel tile. This way, the bottom tiles the plane with my 2D Escher tile, and the top leaves holes where you can now see the figures of the angels and the devils. I’m really happy with the way this turned out!<\/p>\n\n\n\n

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top<\/figcaption><\/figure>\n\n\n\n
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bottom<\/figcaption><\/figure>\n<\/figure>\n\n\n\n

I had some troubles at first printing because the particular shape of the curve made it so that the lofted surface became wider than the tiling slots, so I had to adjust the curve shapes. This is what it looked like before (I just particularly like this image, it contrasts the tiles really well):<\/p>\n\n\n

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The tiling in this image is a little off because I had grid snapping on and didn’t realize, but you can see above that it tiles the plane perfectly. For the actual tiles I printed, I used the offset block to reduce the outer edge as well, so that they could fit to each other. This resulted in some slight gaps, I think as a byproduct of how much I offset, the tiles being small, the geometry, etc. They still fit together very nicely:)<\/p>\n\n\n\n

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Surface Morph<\/h2>\n\n\n
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For this part, I morphed my Escher tiles onto one of my parametric vessels. The code was straightforward from what we accomplished in class. I do have some funkiness going on at the seam but it looks cool.<\/p>\n\n\n\n

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Code<\/h2>\n\n\n\n
LaurenUrendaTiling<\/a>Download<\/a><\/div>\n","protected":false},"excerpt":{"rendered":"

2D Tiling For the Escher Tiles, I was very inspired by Escher’s piece below, Angels and Devils. The original tiling is in hyperbolic space, but I based mine off of this square 2D Euclidean version to the right – I drew a square to emphasize the base tile. I created my two lines in the shapes of where the heads […]<\/p>\n","protected":false},"author":28,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[43,24],"tags":[],"class_list":["post-8036","post","type-post","status-publish","format-standard","hentry","category-tiling_23","category-studentwork23"],"_links":{"self":[{"href":"https:\/\/handandmachine.org\/classes\/computational_fabrication\/wp-json\/wp\/v2\/posts\/8036","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/handandmachine.org\/classes\/computational_fabrication\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/handandmachine.org\/classes\/computational_fabrication\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/handandmachine.org\/classes\/computational_fabrication\/wp-json\/wp\/v2\/users\/28"}],"replies":[{"embeddable":true,"href":"https:\/\/handandmachine.org\/classes\/computational_fabrication\/wp-json\/wp\/v2\/comments?post=8036"}],"version-history":[{"count":8,"href":"https:\/\/handandmachine.org\/classes\/computational_fabrication\/wp-json\/wp\/v2\/posts\/8036\/revisions"}],"predecessor-version":[{"id":8156,"href":"https:\/\/handandmachine.org\/classes\/computational_fabrication\/wp-json\/wp\/v2\/posts\/8036\/revisions\/8156"}],"wp:attachment":[{"href":"https:\/\/handandmachine.org\/classes\/computational_fabrication\/wp-json\/wp\/v2\/media?parent=8036"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/handandmachine.org\/classes\/computational_fabrication\/wp-json\/wp\/v2\/categories?post=8036"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/handandmachine.org\/classes\/computational_fabrication\/wp-json\/wp\/v2\/tags?post=8036"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}