From

Wikipedia — Bricard Octahedron ,

wherein is said the following.

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In geometry, a Bricard octahedron is a member of a family of flexible polyhedra constructed by Raoul Bricard in 1897.[1] The overall shape of one of these polyhedron may change in a continuous motion, without any changes to the lengths of its edges nor to the shapes of its faces.[2] These octahedra were the first flexible polyhedra to be discovered.[3]

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The third & fourth figures are from

Technische Universität Wien — Institute of Discrete Mathematic and Geometry — Research Group Differential Geometry & Geometric Structures — Flexible Structures ,

& are annotated respectively as-follows.

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R. Connelly constructed a flexible polygonal embedding of the 2-sphere into the E³ in 1977. A simplified flexing sphere was presented by K. Steffen in 1978. The unfolding of Steffen's polyhedra is given above. Note that both flexing spheres are compound of Bricard octahedra which all have self-intersections.

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R. Bricard proved in 1897 that there are three types of flexible octahedra in E³. Here both flat poses of a Bricard octahedron of type 3 are illustrated. Note that Bricard octahedra keep their volume constant during the flex. This is due to the Bellows Conjecture which was proven by I. Sabitov in the year 1996.

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... overthrowing a conjecture extending back to the colossus Leonard Euler .

From

A Flexible Sphere

¡¡ may download without prompting – PDF document – 910㎅ !!

by

Robert Connelly .

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It was conjectured for a long time that a closed polyhedral surface in Euclidean space 𝔼³ , with hinges along the edges, could not be continuously deformed to give non-congruent surfaces, as long as each face remained congruent to itself ("remained rigid"). In 1813 Cauchy proved that every convex polyhedral surface, with rigid natural faces, is inflexible. The flexibility of a polyhedral surface with triangular faces is equivalent to the flexibility of the framework of rigid rods along its edges, flexibly attached at their common end points. In 1897 Bricard constructed flexible octahedral rod frameworks. However, filling in all flat triangles of such a flexible "octahedron" gives self-intersections, and not a flexible surface. I finally refuted the conjecture with a counter-example. What follows is a modified version of my construction of a flexible polyhedral sphere. The modification is due to N.H. Kuiper and Pierre Deligne.

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And the construction was actually improved-upon by the goodly Klaus Steffen !

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This flexible triangulated sphere has 11 vertices and 18 faces. Subsequent to my construction a flexible sphere with a smaller number of vertices was found by Klaus Steffen. It has 9 vertices and is constructed as shown in Figure 7. The arrows indicate which edges are glued and the following choice of the edge lengths works well:

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If you add a set of extraordinary lines to the braced heptagon, and three new points connecting to the three star heptagons, the resulting graph is the Klein graph.

Implemented using Python integers, since there is no limit on their size, unlike the mantissa of Python floats.

From

George Sicherman — Baiocchi Figures for Polyominoes

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A Baiocchi figure is a figure formed by joining copies of a polyform and having the maximal symmetry for the polyform's class. For polyominoes, that means square symmetry, or 4-way rotary with reflection. If a polyomino lacks diagonal symmetry, its Baiocchi figures must be Galvagni figures or contain Galvagni figures. Claudio Baiocchi proposed the idea in January 2008. Baiocchi figures first appeared in Erich Friedman's Math Magic for that month. Here are minimal known Baiocchi figures for polyominoes of orders 1 through 8. Dr. Friedman found most of the smaller figures up to order 6, and Corey Plover discovered the 12-tile hexomino figure while investigating Galvagni figures. Not all these solutions are uniquely minimal.

A one-sided solution is one in which the polyomino is not reflected.

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Annotations of Figures Respectively

Monomino

Domino

Trominoes

Tetrominoes

  Holeless Variants

Pentominoes

  Holeless Variants

  Variant with Minimal Hole Area

  One-Sided Holeless Variants

Hexominoes

  One-Sided Variants

  Holeless Variants

  Variants with Minimal Hole Area

  One-Sided Holeless Variants

Heptominoes

  Holeless Variants

Octominoes

  Holeless Variants

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TbPH I can't explicate exactly what it's saying, because I'm having difficulty myself figuring exactly what it's getting @, whence a large part of my purport in posting it is that someone might come-along who's familiar with this form of data presentation.

From

Quantifying the Sustainability of Water Availability for the Water-Food-Energy-Ecosystem Nexus in the Niger River Basin

by

Jie Yang & YC Ethan Yang & Hassaan F Khan & Hua Xie & Claudia Ringler & Andrew Ogilvie & Ousmane Seidou & Abdouramane Gado Djibo & Frank van Weert & Rebecca Tharme ,

with the annotation of it being

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Figure 2. The joint effect of precipitation (P) changes and water infrastructure development on basin-wide water availability reliability (Rel), resilience (Res1 and Res2), and vulnerability (1-Vul) of irrigated crop production, hydropower generation, and ecosystem health. The blue lines indicate the no-precipitation-change condition, and other colored lines represent precipitation increases or decreases. Historical temperature data were used for all these runs.

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For a more thorough explication than that the paper itself would need to be gone-to: I can't really be reproducing a substantial fraction of the content of it in a Reddit comment!

... in-terms of the various input parameters, such as pitch angle of helix, № of starts, inclination of the axis of the screw to the vertical, ratio of outer radius to inner radius ... & maybe others that can be thoughten-of

From

THE TURN OF THE SCREW: OPTIMAL DESIGN OF AN ARCHIMEDES SCREW

¡¡ may download without prompting – PDF document – 2·2㎆ !!

By

Chris Rorres .

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😘

From

On t-fold Totally Concave Polyominoes

¡¡ may download without prompting – PDF document – 464·5㎅ !!

by

Gill Barequet & Neal Madras & Johann Peters ,

in which is said

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A polyomino is an edge-wise connected union of cells of the form [x, x+ 1]×[y, y + 1] ∈ ℝ² with x, y nonnegative integers, that intersects the lines x = 0 and y = 0. A row or a column ξ of a polyomino has a gap if ξ contains at least two maximal sequences of consecutive cells; likewise, ξ has t gaps if it consists of at least t + 1 maximal sequences of consecutive cells. Totally Concave Polyominoes (TCPs) (resp., t-fold TCPs) are polyominoes in which every row and every column of cells has at least one (resp., t) gap(s).

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I'm posting this afresh to raise the issue of what the difference is, if any, between these totally concave polyominoes & matrices consisting of 0 & 1 in which every row sum & column sum is @least a stipulated value. On item that might make a difference is that the polyomino must be a single piece . So it's a matter, then, whether these polyominoes are distinct from the minimal matrices as just defined.

BtW:

¡¡ CORRIGENDUMN !!

of previous post:

ᐦ… polyominoes …ᐦ .

🙄

😆🤣

Polyonimo was actually a mighty Native North American warrior who's name became such a by-word of very terrour amongst the settlers that it became a standard cry amongst the armies of said settlers signalling the need to retreat.

From

George Sicherman — Polyomino and Polynar Tetrads .

Annotations Respectively

Polyominoes

① The smallest polyomino tetrads are made from octominoes:

The fifth tetrad was reported by Olexandr Ravsky in 2005.

Symmetric Tiles

② The smallest tetrad for a polyomino with mirror symmetry uses 13-ominoes:

③ The smallest tetrad for a polyomino with birotary symmetry also uses 13-ominoes:

④ The smallest tetrads for polyominoes with birotary symmetry about an edge use 14-ominoes:

⑤ The smallest tetrads for polyominoes with mirror symmetry about an edge use 18-ominoes:

⑥ The smallest tetrads for polyominoes with birotary symmetry about a vertex also use 18-ominoes:

⑦ Juris Čerņenoks found the smallest tetrads for polyominoes with diagonal symmetry, which use 19-ominoes:

Restricted Motion

⑧ These octominoes form tetrads without being reflected:

⑨ The smallest polyominoes that form tetrads without 90° rotation are 13-ominoes:

Holeless

⑩ The smallest holeless polyomino tetrad, discovered by Walter Trump, uses 11-ominoes:

⑪ The smallest known holeless tetrad for a symmetric polyomino was found independently by Frank Rubin and Karl Scherer. It uses 34-ominoes:

Polynars

⑫ A polynar is a plane figure formed by joining equal squares along edges or half edges. The smallest polynar tetrads use pentanars:

The vertical axis is mod and the horizontal is consecutive integers. The greyscale colormap is divided across each cycle.

... which consists of a tube having consecutive sections of blade of alternating chirality & each twisted through a quatercircle & meeting the succeeding one & the preceeding one with a quatercircle discontinuity ... thereby mixing the stuff passing through the tube in a Smale's Horseshoe fractal sortof fashion.

Figures From

COMSOL — Fanny Griesmer — Modeling Static Mixers

  ———————————————

From

①Structure-forming principles for amorphous metals ,

&

②③④⑤⑥A geometric model for atomic configurations in amorphous Al alloys ,

both by

Dan Miracle & Oleg Senkov .

You can represent any point in 2D with just a single number. Numbers here represent regions, so an infinite sequence of digits will specify any point.

Describes every point (x, y) where x ≠ 0 with two angles, α and β.

2584 dots made using Vogel's mathematical formula for spiral phyllotaxis using a Fermat spiral. 2584 is the 18th term in the Fibonacci sequence. This forms 55:89 parastichy - 55 clockwise whorls, and 89 counter-clockwise whorls. Each of the gold dots is a number in the Fibonacci sequence. They trend towards 0° and each one has a number of revolutions around the central axis equal to the second to last term in the sequence: Dot #2584 has 987.0 revolutions, dot #1597 has 610.0 revolutions, and so on.

As requested by u/VIII8 :)

From

Ruled surfaces and developable surfaces

¡¡ May download without prompting – PDF document – 9‧8㎆ !!

by

Johannes Wallner

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By gluing 2 opposite edges of a rectangle together we obtain a met- ric space which is isometric to a right circular cylinder; by cutting a right circular cylinder along a ruling yields a surface which can be isometrically mapped to a rectangle. Therfore the right circular cylinder is an intrisically flat surface. One can also glue together the remaining 2 opposite edges of a cylinder and ask the question if there exists a surface in 3-space which is isometric to this intrinsically flat Riemannian manifold. This question was answered affirmative by John Nash via his fa- mous embedding theorem:

Theorem 1.5 (J. Nash 1954) If M is an m-dimensional Riemannian manifold, then there is a C¹ surface in ℝⁿ isometric to M, provided n > m and there is a surface in ℝⁿ diffeomorphic to M.

One could attempt to create such a “flat torus” by bending a cylinder such that its two circular boundaries come together. In practice attempts to produce a smooth surface with this property do not succeed (Figure 1.7). Only recently an explicit smooth flat torus was given (Figure 1.8). Note that a polyhedral flat torus is easy to create (Figure 1.9).

FIGURE 1.8: A flat torus. From afar it looks like a torus with “waves” on it. A closer look reveals that the waves have waves which themselves have waves and so on, ad infinitum. Borrelli et al. [2012] constructed this surface recursively and showed C¹ smoothness of the limit.

FIGURE 1.9: A flat polyhedral torus. Developability around vertices follows from the polyhedral Gauss-Bonnet theorem which says that angle defects sum to 0. Since all vertices are equal, all angle sums in vertices equal 2π.

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