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Martin Skrodzki

schemes don't only help finite element methods. In a recent paper, we present a application of creating patterns from said schemes. Furthermore, we extent these patterns to create a puzzle. Here's a thread.🧵1/13 tandfonline.com/doi/full/10.10

Jun 29, 2022, 01:44 PM·Public·Web
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Martin Skrodzki

Computer simulations for, e.g., the weather, solve complex problems on a 2D domain. They mostly do so by splitting the domain into a finite set of smaller and simpler elements on which the simulation can be run fast and efficiently. This uses subdivision schemes. 2/13

An illustration of the subdivision scheme by Loop: On the left an input mesh made of triangles; into the input mesh, the scheme inserts new vertices as edge midpoints; then, these new vertices are connected to triangles, creating smaller ones than those from the input; finally, the input is smoothed to make the mesh more regular.
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Martin Skrodzki

If the output domain should be triangles, there is the Loop subdivision (see image from previous tweet), the butterfly🦋 scheme, or the √3 scheme. For output made of quadrilaterals, use mid-edge, Doo-Sabin, or Catmull-Clark. Irregular input gives irregular output, though. 3/13

An illustration of the subdivision scheme by Catmull-Clark: On the left an input mesh made of triangles, quadrilaterals and a pentagon; into the input mesh, the scheme inserts new vertices as edge and face midpoints and connects all face midpoints with the incident edge midpoints. Irregularities of the input mesh are retained during the refinement steps: The subdivision scheme of Catmull-Clark mainly creates vertices of degree four, but has vertices of different degree wherever the input mesh has a non-quadrilateral face.
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Martin Skrodzki

None of these schemes create pentagonal faces. We discuss a procedure where, by replacing each edge by a Z-shaped triplet of edges and connecting their vertices to face-midpoints, the output mesh becomes a mesh purely comprised of pentagonal faces. 4/13

From left to right: Illustration of a Z-triplet with its two enclosed 2π/3 angles, where the original line that is replaced by the Z-triplet is shown dashed; Also shown is the input mesh M0 with its faces colored blue and a boundary to the right; On this input mesh, each edge is replaced by a Z-shaped triplet of edges, such that one of the two inner vertices is on the one and the other is on the other side of the original edge; Finally, the inner vertices of the Z-Triplet are connected to the face midpoints of those face in which they lie.
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Martin Skrodzki

Applying this subdivision leads to non-convex pentagons. Thus, we perform a smoothing step, moving vertices towards the barycenter of their original faces. Whether or not this operation keeps the faces convex for all inputs and arbitrary many subdivisions is an open problem. 5/13

Illustration of the smoothing: white vertices are moved to blue ones. For each face f of the mesh M, as obtained after the subdivision step as described above, its barycenter bf is computed. Then, for each inner vertex v, its new position is determined as the average of the face barycenters bf of those faces f to which v is incident. After determining the new positions, all vertices are updated at once.
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Martin Skrodzki

A first object arises when considering repeated pentagon snub subdivisions of an initial, regular pentagon. In this case, the boundary curve subdivides to a curve. Whether or not inner curves form fractals given the smoothing is an open problem. 6/13

An illustration of a regular pentagon, subdivided by the snub pentagon subdivision scheme five times. The outer curve of the resulting mesh is drawn bold, highlighting it as the fractal curve.
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Martin Skrodzki

As mentioned, repeatedly applying the pentagon snub subdivision scheme to a regular pentagon does not only yield a curve on the outer boundary, but also fractal-like curves inside the mesh. These arise both from edges as well as from the remaining faces. 7/13

Illustration of repeated pentagon snub subdivision refinement applied to a regular pentagon. The image shows the 8th refinement step. Shown are the edges resulting from the corresponding refinement step. The left over white faces shine through the black edges in a fractal-like pattern. But also the black edges form fractal-like patterns, as a zoom-in window reveals.
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Martin Skrodzki

From the subdivision, we can create a by removing each middle edge of the introduced Z-triplets, i.e., gluing two pentagons together. A strand of the weaving is given by those tiles whose tips are connected by a now absent middle edge of a Z-triplet. 8/13

Illustration of how to create a strand of the weaving pattern (in four images). The first image shows a regular square grid, once subdivided into pentagons via the snub pentagon subdivision scheme. The second image highlights the middle edges of the introduced Z-triplets of the subdivision scheme. In the third image, all other edges are shown bold, while the middle Z-triplet edges are very thin, as they are to be deleted. The final image then shows three tiles, each comprised of two glued-together pentagons, while the tiles are connected in a straight line by middle edges of the Z-triplets.
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Martin Skrodzki

An interesting phenomenon arises when observing a strand of the under a subdivision operation. The strand splits into two new strands, which are intertwined. This also holds for an additional operation, giving four strands in total that form pairs. 9/13

Illustration of strand splitting. Starting with the fourth image from the previous thread, a subdivision step is applied. This yields two smaller strands that go on top and below each other. Applying the subdivision another time yields four strands that form two pairs. Within each pair, the strands are going on top and below each other. But additionally, the pairs also go on top and below each other.
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Martin Skrodzki

The described now leads to a activity: Given the square or the regular pentagon and some simple paper cutouts, we can build the weaving by hand. Find the necessary templates attached to this tweet and give it a try! 10/13

Template for building the three times subdivided pentagon.
Template for building the twice subdivided square grid oneself.
Picture of a paper version of a twice subdivided square grid.
Picture of a paper version of a pentagon with adjacent pentagons on each side, which has been subdivided three times.
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Martin Skrodzki

In the fifth and second-to-last chapter of the paper, we discuss how the can be generalized to other subdivision schemes, such as the aforementioned scheme by Loop. Here, subdividing strands gives two parallel, not intertwined strands. 11/13

From left to right, three images illustrating the weaving patterns from the subdivision scheme by Loop. In the first image, consider a two-coloring of the triangular input mesh such that each face has exactly one vertex of the one color and exactly two vertices of the other color. Now, remove dashed edges between vertices both colored by the same color and thereby glue triangles accordingly. Final illustration shows that applying the subdivision splits a strand into two parallel ones.
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Martin Skrodzki

For our final piece, we realized one of the subdivisions as a , much in the spirit of @nervous_system. The two sides of the puzzle illustrate the strands by showing them as line segments or rather by coloring in the glued mesh faces. 12/13

Photo of the other side of the jigsaw puzzle, showing the strands as colored-in faces of the underlying mesh.
Photo of the jigsaw puzzle, showing the strands as colored lines twisting over the surface.
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Martin Skrodzki

Our work is in @MathsArts, thanks to the support by @tudelft. Thanks to my co-authors for a great collaboration, to the editors @Gelada and @henryseg, and to @dfg_public for supporting my research conducted at @TUDelft_CGV. Full article: tandfonline.com/doi/full/10.10 13/13

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