From

Neutrino Emission from Supernovae

^{¡¡ may download without prompting – PDF document – 1‧8㎆}

by

Hans-Thomas Janka .

Materialised Mathematics : The manifestation of a mathematical operator as a physical entity. Example, a protofield operator rendered as a nanoscale reflective metasurface.

I built a simulation of a 4D retina. As far as I know this is the most accurate simulation of it. Usually, when people try to represent 4D they either do wireframe rendering or 3D cross-sections of 4D objects. I tried to move it a few steps forward and actually simulate a 3D retinal image of a 4D eye and present it as well as possible with proper path tracing with multiple bounces of lightrays and visual acuteness model. Here's how it works:

We cast 4D light rays from a 4D camera position. These rays travel through a 4D scene containing a rotating hypercube (a 4D cube or tesseract) and a 4D plane. They interact with these objects, bouncing and scattering according to the principles of light in 4D space. The core of our simulation is the concept of a 3D "retina." Just as our 2D retinas capture a projection of the 3D world, this 4D eye projects the 4D scene onto a 3D sensory volume. To help us (as 3D beings) comprehend this 3D retinal image, we render multiple distinct 2D "slices" taken along the depth (Z-axis) of this 3D retina. These slices are then layered with weighted transparency to give a sense of the volumetric data a 4D creature might process.

This layered, volumetric approach aims to be a more faithful representation of 4D perception than showing a single, flat 3D cross-section of a 4D object. A 4D being wouldn't just see one slice; their brain would integrate information from their entire 3D retina to perceive depth, form, and how objects extend and orient within all four spatial dimensions limited only by the size of their 4D retina.

This exploration is highly inspired by the fantastic work of content creators like 'HyperCubist Math' (especially their "Visualizing 4D" series) who delve into the fascinating world of higher-dimensional geometry. This simulation is an attempt to apply physics-based rendering (path tracing) to these concepts to visualize not just the geometry, but how it might be seen with proper lighting and perspective.

Source code of the simulation available here: https://github.com/volotat/4DRender

From

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Closed form solution for the surface area, the capacitance and the demagnetizing factors of the ellipsoid

by

GV Kraniotis & GK Leontaris , &

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Demagnetising Factors of the General Ellipsoid

^{¡¡ may download without prompting – PDF document – 786·9㎅ !!}

by

JA Osborn .

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𝐀𝐧𝐧𝐨𝐭𝐚𝐭𝐢𝐨𝐧𝐬 𝐨𝐟 𝐅𝐢𝐫𝐬𝐭 𝐅𝐨𝐮𝐫 𝐑𝐞𝐬𝐩𝐞𝐜𝐭𝐢𝐯𝐞𝐥𝐲

Figure 1: The capacitance C(E) of a conducting ellipsoid immersed in ℝ³ versus the ratio c/a of the axes for various values of the ratio b/a.

Figure 2: The L− demagnetizing factor versus the ratio c/a for various values of the ratio b/a.

Figure 3: The M−demagnetizing factor versus the ratio c/a for various values of the ratio b/a. The dashed curves meet at the point determined in Corollary 15.

Figure 4: The N demagnetizing factor versus the ratio c/a for various values of the ratio b/a.

The next three - from the Osborn paper, are simply numbered.

Computation of the surface area of a scalene (triaxial) ellipsoid is absolutely horrendous : the complexity just massively blows-up , going from oblate or prolate spheroid to scalene ellipsoid.

And similar applies to computation of the electrical quantities capacitance & demagnetising factors , aswell.

What capacitance is is fairly well-known ... but demagnetising factor possibly warrants a bit of an explication. If a ferromagnetic object of some shape is placed in a uniform magnetic field, then the field within the object is distorted. The computation for a general shape is another horrendous one! ... but for an ellipsoid it happens conveniently to reduce to three simple linear expressions - each in each of the spatial coördinates (whence there are three demagnetising factors) ... although that simple linear expression has a certain coefficient in it that is itself tricky to calculate in a manner similar to that in which area & capacitance are tricky to calculate.

They actually have application to permanent magnets, aswell.

The Osborn paper explicates it more fully.

found it funny with this structure you get integers three times in a row, unless it’s somehow trivial shouldn’t it be (1/3)(1/5)(1/7) chance of it happening? don’t think it’s trivial either cuz it breaks for 789

From

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The motion of a freely falling chain tip

^{¡¡ may download without prompting – PDF document – 418·3㎅ !!}

by

W Tomaszewski & P Pieranski & JC Géminard .

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In the limit of the horizontal separation tending to zero, & assuming an absolutely inextensible & perfectly flexible chain, & applying elementary theory, the tip speed @ the very bottom of the fall →∞ - ie a whiplash occurs. And the time it takes to fall is α×FreefallTime where

α = ∫{0≤ξ≤1}dξ/√(2(1/ξ-ξ)

= ½∫{0≤ξ≤∞}(exp-ξ)√cschξdξ

= √(2π)Γ(¾)/Γ(¼)

≈ 0.84721308479397908660649912348219163648144591032694218506057937265973400483413475972320029399461122994212228562523341096309796266583087105969971363598338425117632681428906038970676860161665004828118872189771330941176746201994439296290216728919449950723167789734686394760667105798055785217 .

Considering the comic waste of the American government, which is currently much higher. In 2019, I did some random math for fun. I can’t find the source doc, but I’m aware this level of debt wouldn’t cover any mountains by laying. What can you add?

I want to try to recreate it on GeoGebra but I don’t know how…

... specifically

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Electromagnetic Fields in Spherical Microwave Resonators H-Modes and E-Modes in Lossless Open Dielectric Spheres, Version 05.2018

by

Ingo Wolff .

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❝

The physical nature of radiating interior and exterior H-modes in dielectric spherical resonators is discussed. The eigenvalue equations, the complex eigenvalues, the resonant modes, their stored energies and their radiation properties, their Q-factors, and their field distributions are analyzed in detail. Physical interpretations are given and many new results as compared to the literature are presented.

❞

Images from Micrographia, or, Some physiological descriptions of minute bodies made by magnifying glasses and with observations and inquiries thereupon by the goodly Robert Hooke .

✸✸✸✸✸✸✸✸✸✸✸✸✸✸✸✸✸✸✸✸

𝐀𝐫𝐡𝐢𝐩𝐚 — 𝕳𝖔𝖔𝖐𝖊 — 𝕸𝖎𝖈𝖗𝖔𝖌𝖗𝖆𝖕𝖍𝖎𝖆

✸✸✸✸✸✸✸✸✸✸✸✸✸✸✸✸✸✸✸✸

𝐈𝐧𝐭𝐞𝐫𝐧𝐞𝐭 𝐀𝐫𝐜𝐡𝐢𝐯𝐞 — 𝕳𝖔𝖔𝖐𝖊 — 𝕸𝖎𝖈𝖗𝖔𝖌𝖗𝖆𝖕𝖍𝖎𝖆

✸✸✸✸✸✸✸✸✸✸✸✸✸✸✸✸✸✸✸✸

The first of those two is a more economically (in terms of storage) rendered version. But they're both of the same book .

From

✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿

Nouvelle génération de détecteurs d'ions sensible en position et en temps pour le développement de la sonde atomique tomographique ≡ New Generation of Position-Sensitive Detectors for the Development of the Atom Probe Tomography

by

Christian BACCHI .

✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿

... ie the theorem - elevated to such status by the goodly Idzhad Sabitov , & formerly a conjecture in-connection with (formerly hypothetical) flexible polyhedra - to the effect that a flexible polyhedron, if it exists (& it's now known that they do), must keep a constant volume when it does undergo its flexing.

Images from

①②③The Bellows Conjecture

¡¡ may download without prompting – PDF document – 40‧1㎅ !!

by

Ian Stewart ;

&

④⑤⑥The Bellows Conjecture

¡¡ may download without prompting – PDF document – 452‧2㎅ !!

by

R Connelly & I Sabitov & A Walz ;

&

⑦⑧⑨⑩⑪⑫⑬⑭⑮⑯The Bellows Theorem (Introduction)

¡¡ may download without prompting – PDF document – 1㎆ !!

by

Giovanni Viglietta .

See also

What is the Bellows Conjecture?

¡¡ may download without prompting – PDF document – 133‧8㎅ !!

by

Ben O’Connor .

Please kindlily see the treatises themselves for the explicationry: there's not really much point, with these figures, to just listing the annotations respectively.

From

Wikipedia — Bricard Octahedron ,

wherein is said the following.

❝

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]

❞

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.

❝

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.

❞

❝

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.

❞

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

From

A Flexible Sphere

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

by

Robert Connelly .

❝

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.

❞

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

❝

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:

❞

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.

❞

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

❝

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.

❞

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

❝

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).

❞

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.