u/BadJimo

A few days ago someone asked for the dimensions of a Sasha's Hexacontahexahedron - D66 dice, and then deleted the post.

Anyway, I have made what I think is the requested Hexacontahexahedron.

Illustrated on Desmos

The polyhedron is more complicated than it first appears. There are 6 hexagons and 60 irregular pentagons. However, the hexagons are not quite regular (two of the angles and two sides and slightly different to the other four angles and sides respectively). There are three types of similar looking pentagons: 12 of one type (green) that is symmetrical, and two other types (yellow and orange) that are not symmetrical (24 of each).

An AI recently found a counterexample to the Jacobian conjecture.

𝑓(𝑥,𝑦,𝑧)=((1+𝑥𝑦)3𝑧+𝑦2(1+𝑥𝑦)(4+3𝑥𝑦),𝑦+3𝑥(1+𝑥𝑦)2𝑧+3𝑥𝑦2(4+3𝑥𝑦),2𝑥−3𝑥2𝑦−𝑥3𝑧)

I wondered if the counterexample to the Jacobian conjecture could be visualised.
Here's what I came up with:

https://www.desmos.com/3d/cqjyedoauh

The red curve (a simple 3D spiral on the surface of a sphere) is the input of the Jacobian conjecture counterexample, the blue curve is the transformed output.

https://arxiv.org/abs/2510.24491

From the abstract:

>We provide arguments indicating that the semiclassical Einstein equations follow from quantum relative entropy and its proportionality to an area variation. Using modular theory, we establish that the relative entropy between the vacuum state and coherent excitations of a scalar quantum field on a bifurcate Killing horizon is given by the energy flux across the horizon. 

[Sabine Hossenfelder has positive view of this paper](https://youtu.be/IxmTqIkToTg) which Gemini has summarized:

. The Core Mystery: Einstein's Rulebook

​[00:00] – Introduction to Entropic Gravity: Sabine explains a potentially groundbreaking realization in theoretical physics: gravity might not be a fundamental force like electromagnetism, but rather an emergent property driven by entropy (the measure of disorder or information in a system).

​[00:29] – The Limits of General Relativity: Einstein’s equations beautifully describe how matter and energy warp spacetime (causing paths to bend and clocks to tick differently), but they function as a rulebook rather than an explanation. They don't explain the underlying mechanism for why matter curves spacetime in exactly this way.

​2. The Black Hole Connection

​[01:11] – Bekenstein and Hawking's Discovery: The thermodynamic approach to gravity isn't brand new. It builds on work from 50 years ago when Jacob Bekenstein and Stephen Hawking discovered that black holes possess temperature and entropy.

​[01:35] – Area and Information: Mathematically, a black hole's entropy is directly proportional to its surface area (measured in fundamental Planck areas). This link hinted that gravity and thermodynamics might be deeply intertwined on a foundational level.

3. Expanding Thermodynamics Everywhere

​[01:57] – Ted Jacobson’s 1995 Equation: Physicist Ted Jacobson showed that this thermodynamic connection wasn't unique to black holes. By applying the mathematics of horizons everywhere in space and tracking how energy crossing a boundary changes entropy, he was able to mathematically derive Einstein’s equations from scratch.

​[02:36] – The New Milestone (Quantum Fields): The problem with Jacobson's earlier derivation was determining exactly what was shifting in entropy. The new paper highlighted in this video solves this by showing the math works perfectly using the relative entropy of quantum fields (the systems describing all Standard Model particles).

​[02:58] – Measuring Quantum States: The authors calculated the mathematical difference (relative entropy) between a pure vacuum state and a vacuum state containing a minute amount of matter or energy. This difference directly correlates to energy flow, tying the information changes back to spacetime geometric modifications.

​4. Gravity as an "Emergent" Illusion

​[03:41] – Why Gravity Isn't a Quantum Theory: This derivation naturally recovers Einstein's classic, non-quantum equations. If this paper holds up, it elegantly explains why physicists have struggled to find a "quantum gravity" theory (like gravitons or string theory variants)—because gravity itself isn't a fundamental quantum force. It is a macro-scale bulk property that emerges out of quantum information, much like how heat and pressure emerge from the collective motion of microscopic gas molecules.

​[03:55] – The Remaining Question: Sabine rates the paper highly (a low 1 out of 10 on her "bullshit meter"), but notes a glaring omission that entropic gravity theorists have yet to solve: if gravity is an emergent property, what fundamental building blocks is it emerging *from*?

A cevian is a line that connects the vertex of a tetrahedron to somewhere on the opposite face of a tetrahedron. The somewhere is usually a mathematically interesting point such as the centre of the face. Now something interesting about triangles is that there are many different types of centre of a triangle. Here (not made by me; just found in a Google search) is a wonderful Desmos graph that shows the 10 most interesting/useful centres of a triangle. I will use the Gergonne triangle centre in this project (which I wasn't aware of until today).

Mostly, the cevians of an irregular tetrahedron do not intersect. I wanted to make a tetrahedron where the cevians do intersect.

Apparently if the tetrahedron is an inspherical tetrahedron then the cevians that extend from the Gergonne centre of each face of the tetrahedron to the opposite vertex do intersect.

It turns out that if you make a tetrahedron with a sphere centred at each vertex and the spheres are tangent to each other (as I made recently in another project) then this is the right kind of tetrahedron. Yay.

Anyway, here is a link to the graph (the green dots are at the Gergonne centres):

https://www.desmos.com/3d/ngyznuch9u

I recently discovered the marvelous Van Aubel's theorem. If you have a quadrilateral with a square on each edge, then the line segments from centres of opposite squares will be perpendicular and the same length.

I wondered if there was a 3D extension of Van Aubel's theorem. I couldn't find one, so I made one. My first thought was a cube at every face of a hexahedron, but obviously that won't work. So instead I thought tangent spheres are kind of similar.

Start with 6 spheres, each sphere tangent to 3 adjacent spheres. That is, spheres at the vertices of a hexahedron (think a distorted cube).

Then add a sphere tangent to each group of 4 spheres. The connect the centres of these opposite spheres to make three lines, and voilà, the three lines are orthogonal and two are the same length... under some conditions. Specifically, only when the 'centre of inversion' is only modified in one axis. I used the spherical inversion technique I mentioned in my previous post, but this time using a cube.

When the 'centre of inversion' is modified in two axes, you still have two of the lines being orthogonal.

I've called it Van Baubel's theorem because, well, I think you can figure it out.

https://www.desmos.com/3d/letm5p0sfh

Four tangent (kissing) spheres.

https://www.desmos.com/3d/0fw8xdiedu

I started with a regular tetrahedron with spheres of the same size at the vertices and then applied a 3D Möbius transformation known as Spherical Inversion.

​A spherical inversion turns 3D space inside-out through a "lens" (a sphere of inversion). Things close to the lens get blown up and pushed far away, while things far away get shrunk and pulled inside.

​An important property of this transformation is that it preserves tangency. If two spheres kiss before the inversion, they will kiss after the inversion.

To change the sizes of the sphere you move the "Center of Inversion" (using the sliders o,p,q).

I realized a corner cube reflector only requires a single line to define the ray of light.

https://www.desmos.com/3d/3vb2nni5j4