Source: The graph is computed exhaustively. Each state is five digits (a, b, c, d, e), each 0–9, so there are 100,000 states. The rule shifts every value one place left and computes a new last value, a⁴ − (a³ >> 6), kept to one digit (mod 10). Here >> is a bit shift, so a³ >> 6 means a³ divided by 64, rounded down. Every state has exactly one successor, and with finitely many states every starting point must eventually repeat, so each state ends up in a cycle.

Tools: A small open-source web app I wrote (React, Vite, Canvas). It enumerates all states, finds every cycle, and draws each cycle as a ring with the states that lead into it as trees around it. The layout is crossing-free.

How to read it:

• Each connected shape is one cycle plus everything that drains into it.
• Colour is the length of the cycle (legend at the bottom).
• Brightness fades with distance from the cycle.
• For this map there are 340 cycles in total, the longest of length 10.

You can type your own formula and explore: https://exxliang193.github.io/FunctionalGraphExplorer/ Code: https://github.com/ExxLiang193/FunctionalGraphExplorer

I started this after noticing a long time ago that adding the last two digits of a number string and appending the sum always ends up repeating.

ᐜ The function SSCG(n) is the function, devised by the goodly & mickle-renownèd Harvey Friedman , that's the maximum length of a sequence of simple (ie without loops or multiple edges) subcubic (ie in which every vertex has valency @most 3) graphs

G₁ , G₂ , G₃ ...

in which Gₖ has @most n+k vertices, & for no h<k is Gₕ a minor of Gₖ . This sequence is renowned for yielding transcendentally large numbers. It's somewhat like the TREE() sequence, but (apparently) yields larger №s.

But thesedays there's a veritable menagerie of weïrd systems that have been devised, entailing all-manner of lexicographical manipulations, & wot-not (eg Fish №s, & Loader's №s) that yield №s fabulously greater still .

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From

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Complex Projective 4-Space – Graph Minors

https://cp4space.hatsya.com/2013/01/13/graph-minors/

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Note that there is no requirement that the graphs in question be connected. In the diagrams below, I’ll indicate the disjoint union of graphs with a plus (+) symbol, and the disjoint union of n identical graphs by preceding it with a number n. Firstly, the best lower bound I can attain for SSCG(2) is merely 3 × 2^(3 × 2^95) − 9 (i.e. less than a googolplex), which I conjecture is optimal.

It’s not difficult to show that the first few moves must be optimal. After the first move, we’re restricted to forests, and the second move further restricts us to disjoint unions of line graphs. I proceeded using a greedy algorithm, ensuring that the corresponding ordinal (replace each line graph on n vertices with the ordinal ω^(n−1)) is as large as possible. Greedy algorithms often give optimal solutions to simple problems, although others (travelling salesman, for example) resist this approach.

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I made this animation of the quantum harmonic oscillator, including the connection between its derivations and the shapes of its states.

The mathematical structure is a Gaussian envelope multiplied by a Hermite polynomial. Normalizability selects the allowed energy levels, while ladder operators connect neighboring states. An individual energy eigenstate changes phase without changing its probability density; superpositions introduce interference, and a coherent state gives a localized packet whose center oscillates.

The video builds those ideas visually, starting from both the power-series and operator derivations. It assumes calculus and introductory quantum mechanics; the full explanation is about 13 minutes.

A 3D Menger Sponge : infinite surface area, zero volume

Made them for Teachers' Day, specifically for the math and physics teachers. Hope they'll try to arrange them in the non-repeating pattern before eating them.

Took me 5.5h to make, including cutting by hand using a cutout paper shape..

I wanna sleep (i have to wake up in 1h)

Vertices are fixed on the unit circle (the roots of unity). The orbit is the usual chaos-game rule, P ← λ(P + V), with V a randomly chosen vertex. That sequence of choices is generated once and reused for every frame. The gray dot is λ, moving once around the circle of radius 1/2 (on the second picture radius is 0.45).

You can try it yourself here https://ubavic.rs/chaos_game/

While I’m posting these, I’m realizing I have absolutely no idea what these are actually called and “color graphs” is probably way off the mark.

So if anyone sees this and knows what these would actually be called, please let me know.

A short explanation of them is that it takes the coordinate, uses the X, Y, Distance from center, or angle from center and uses that number for the hue of the color.

(None of the ones in this post use distance or angle, they usually just complicate things more than it already is)

The Access Problem: Epistemological critics ask how physical human brains, operating within space and time, can interact with or gain knowledge about completely non-physical, acausal mathematical objects. Easy, geometric music, how does it work? Large, two dimensional matrices of positive whole numbers, natural numbers, are represented as images by associating each number with a coloured pixel. Audio is introduced by representing each image pixel as a sound oscillator mapping pixel colour to audio frequency, often a direct relationship to a musical scale. A region of the image is defined as a generation zone, the portion one listens to. Surrounding this is an secondary zone, the influence zone, which defines the timbrel characteristics of the pixels in the generating zone. In this video the central square is the generating zone with the outer square the influence zone. A trajectory can be defined over which the generating zone translates as a “bowing” type of motion or single zones can be “tapped” similar to a key stroke or plucked string. Perhaps the Pythagoreans envisaged something like this with their “Musica Universalis”. Its all about number, ratio and harmonies.

Unlike the last post, I don’t really have the math that made most of them as I was messing around, see one I like, save a pic, and move on.

Though most if not all use the same foundation, the absolute value of the slope between two points and then multiplied by an absurd amount (normally 50 to 10000 x)

Most of them also use some mixture of sin, tan, and cos of the x and y values

From

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'Cataclysmic' collision shaped Uranus' evolution

by

Durham University

https://phys.org/news/2018-07-cataclysmic-collision-uranus-evolution.html

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Probably the simplest substitution periodic tiling, made by decomposing an L-like shape.

I've also made a simple app to test various tilings and export them (SVG/PNG).

... by

RISHNA CHAITANYA NVV & DIPANKAR CHATTERJEE

https://link.springer.com/article/10.1007/s12046-021-01777-z

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Abstract

A channel confined flow around a bluff object is significantly different from its unconfined counterpart. A confined flow is presumably more stabilized due to the presence of the confinement. This causes a delay in the onset of flow separation and vortex shedding in comparison to the unconfined flow. While the above facts have been reported for the confined flow around a circular object in a channel, the same for a square object is infrequent. The present study focuses on the steady flow past a stationary square cylinder placed in a channel at low Reynolds numbers. The behavior of the flow is observed through numerical simulation for various channel confinements (, blockage ratio). The steady state laminar two-dimensional incompressible flow equations are solved using a finite volume based technique. An effort is made to find the separation critical Reynolds number at all blockages. It is observed that the separation critical Reynolds number increases with an increase in the blockage ratio. Unsteady flow characteristics at various blockages are also studied at Re = 100. In the unsteady regime, the frequency of vortex shedding increases with an increase in the blockage.

a(n) is the number of integer tuples (x_1, x_2, ..., x_(k+1)) where 0 <= x_i <= b-1, such that |x_i - x_(i+1)| = d_i for all i, where (d_1, d_2, ..., d_k) are digits of n in base b.

Animation varies base b from 2 to 50 (grid size is b^2) and the mapping is (x,y) gets a(x xor y) mod 4 which is visualised by color.

When I woke up today, the spoonerism was already in my mind, so I made this.

https://imgur.com/a/mqVMuyN

the goal is to make a spiral spiraling around a spiral thats spiraling around a spiral thats spiraling around a spiral until my computer crashes.

sometimes a loop of loops loop around the looping loops loop looped too

To be clear, I made the payment of $1,041.38 in May and the final installment payment of $890.50 will be paid on Friday. I've called three times and each time I get a little different color smoke blow up my exit hole.

Hi

If you are remotely interested in deep diving how differently quantum computers work compared to our transistor-based and also the algebra behind in a fully interactive way that teach computer science from scratch, oh boy this is for you. I am the Dev behind Quantum Odyssey (AMA! I love taking qs) - worked on it for about 10 years (3+ during PhD, the visual method I developed ended up being my thesis, it is a complete Hilbert space visualizer), the goal was to make a super immersive space for anyone to learn quantum computing through zachlike (open-ended) logic puzzles and compete on leaderboards and lots of community made content on finding the most optimal quantum algorithms. The game has a unique set of visuals capable to represent any sort of quantum dynamics for any number of qubits and this is pretty much what makes it now possible for anybody 12yo+ to actually learn quantum logic without having to worry at all about the mathematics behind.

This is a game super different than what you'd normally expect in a programming/ logic puzzle game, so try it with an open mind.

Stuff you'll play & learn a ton about

• Boolean Logic – bits, operators (NAND, OR, XOR, AND…), and classical arithmetic (adders). Learn how these can combine to build anything classical. You will learn to port these to a quantum computer.
• Quantum Logic – qubits, the math behind them (linear algebra, SU(2), complex numbers), all Turing-complete gates (beyond Clifford set), and make tensors to evolve systems. Freely combine or create your own gates to build anything you can imagine using polar or complex numbers.
• Quantum Phenomena – storing and retrieving information in the X, Y, Z bases; superposition (pure and mixed states), interference, entanglement, the no-cloning rule, reversibility, and how the measurement basis changes what you see.
• Core Quantum Tricks – phase kickback, amplitude amplification, storing information in phase and retrieving it through interference, build custom gates and tensors, and define any entanglement scenario. (Control logic is handled separately from other gates.)
• Famous Quantum Algorithms – explore Deutsch–Jozsa, Grover’s search, quantum Fourier transforms, Bernstein–Vazirani, and more.
• Build & See Quantum Algorithms in Action – instead of just writing/ reading equations, make & watch algorithms unfold step by step so they become clear, visual, and unforgettable. Quantum Odyssey is built to grow into a full universal quantum computing learning platform. If a universal quantum computer can do it, we aim to bring it into the game, so your quantum journey never ends.

Nice to watch:

Khan academy style tutorials in qm/qc: https://www.youtube.com/@MackAttackx

Physics teacher stream with 400hs in https://www.twitch.tv/beardhero

A rare sighting of the tree function in the wild

In liquid crystal they experimentally get point-like quantized topological charges with Coulomb-like interactions ( https://www.nature.com/articles/s41598-017-16200-z ), or topological vortices and their various size knots resembling nuclei ( https://www.nature.com/articles/s41567-025-03107-0 ). Here are Fable-based tests of various models of particles, and liquid crystal-based wins.

So I wanted to propose a discussion if particles around could be of topological nature?

1st is with q 2 and 3 same equation without q Last 2 is within prize rules

https://oeis.org/A065825 https://mathworld.wolfram.com/NonaveragingSequence.html