Vitaliy Kaurov

Curious. Entangled. Chief Editor @ Wolfram Staff Picks. Physicist.

[Vitaliy Kaurov @superflow]

Fantastic. Mostly 𝐬𝐢𝐧(𝐱). Name it... Fireeel? Remix in #Wolfram Mathematica. Full code below.

x = Range[0., 9999]; k = 4 Cos[x/21];
e = x/1880 - 20; d = Sqrt[k^2 + e^2];
m = UnitStep[k^2 - 15];

Manipulate[With[{
q = 3 Sin[2 k]+.3/k+k*Sin[x/4465](9+2*Sin[14*e-3*d+2*t])},
Graphics[{
Blend[{White, Red}, Sin[t]^2], Opacity[.5], PointSize[.01],
Point@Pick[#, m, 1],
White, Opacity[.75], PointSize[.0025],
Point@Pick[#, m, 0]}&@
Transpose@{q+50*Cos[d-t]+200, 875-q*Sin[d-t]-39*d},
PlotRange -> {{100, 300}, {75, 320}}, Background -> Black]],
{t, 0, 2 Pi}]

[ア @yuruyurau]

a=(x,y,d=mag(k=4*cos(x/21),e=y/8-20))=>circle((q=3*sin(k*2)+.3/k+sin(y/19)*k*(9+2*sin(e*14-d*3+t*2)))+50*cos(c=d-t)+200,q*sin(c)+d*39-475,k*k>15?2:1)
t=0,draw=$=>{t||createCanvas(w=400,w);background(9).noStroke().fill(w,116);for(t+=PI/240,i=1e4;i--;)a(i,i/235)}#つぶやきProcessing

[Vitaliy Kaurov @superflow]

𝐌𝐚𝐫𝐭𝐢𝐚𝐧 𝐃𝐢𝐜𝐡𝐨𝐭𝐨𝐦𝐲 got two mechanisms: a giant impact and internal magma flow. Over four billion years ago, a Pluto-sized object likely smashed into the northern hemisphere, blasting away the crust to leave a massive basin. Alternatively, a single giant plume of hot magma inside the planet pushed upward, thickening the southern crust. The most accepted theory combines both: the colossal northern asteroid strike generated a thermal shockwave that forced the planet's internal magma to rise and build the elevated southern highlands. Here is also static high resolution poster. Full screen video recommended for elevation texture. Full animation code:
https://t.co/bawMvrdA9R

[Vitaliy Kaurov @superflow]

Mars is lopsided - called 𝐌𝐚𝐫𝐭𝐢𝐚𝐧 𝐃𝐢𝐜𝐡𝐨𝐭𝐨𝐦𝐲. Oceans in North, continents in south - if ones imagines water on Mars at the same 71% surface area as water on Earth.

Mars topography is like Yin-Yang symbol: its highest spot Olympus Mons (solar system tallest mountain) is in lowlands North, and its lowest spot Hellas Planitia (one of the largest craters in solar system) is in highlands South.

Amazingly simple programing trick gives minimal Wolfram code below to visualize all this fascinating and unique Mars topography.

The trick: sample Mars geo-elevation uniformly from equal-area map projection and then quantile points at 71% - the value you get splits lowlands and highlands. If you flood lowlands to that value it yields 71% global water surface.

🔴 WOLFRAM CODE:

elSamp = Flatten @ QuantityMagnitude @ GeoElevationData[
GeoProjection -> "CylindricalEqualArea",
GeoZoomLevel -> 1, GeoRange -> "World", GeoModel -> "Mars"
];

tHeight = Rescale[Quantile[elSamp, 0.71], MinMax[elSamp]];

GeoGraphics[
GeoModel -> "Mars", GeoRange -> "World", GeoProjection -> "VanDerGrinten",
GeoBackground -> GeoStyling["ReliefMap",
ColorFunction -> (If[# < tHeight, StandardBlue, StandardOrange] &)
]
]

[Vitaliy Kaurov @superflow]

Wolfram one-liner code verification:
In[]:= Complement[Alphabet[], Characters["" <> CommonName@ElementData[]]]
Out[]= {"j", "q"}
This dataset only checks current official names. "Q" is absent today, but it was used historically in temporary IUPAC placeholders (e.g., ununquadium), leaving "J" as the only letter to never appear in any capacity.

[Vitaliy Kaurov @superflow]

@Flexi23 Keen eye. Someone mentioned to me Cambrian/Ediacaran fossil aesthetics.

[Vitaliy Kaurov @superflow]

Fantastic. Mostly 𝐬𝐢𝐧(𝐱). Name it... Fireeel? Remix in #Wolfram Mathematica. Full code below.

x = Range[0., 9999]; k = 4 Cos[x/21];
e = x/1880 - 20; d = Sqrt[k^2 + e^2];
m = UnitStep[k^2 - 15];

Manipulate[With[{
q = 3 Sin[2 k]+.3/k+k*Sin[x/4465](9+2*Sin[14*e-3*d+2*t])},
Graphics[{
Blend[{White, Red}, Sin[t]^2], Opacity[.5], PointSize[.01],
Point@Pick[#, m, 1],
White, Opacity[.75], PointSize[.0025],
Point@Pick[#, m, 0]}&@
Transpose@{q+50*Cos[d-t]+200, 875-q*Sin[d-t]-39*d},
PlotRange -> {{100, 300}, {75, 320}}, Background -> Black]],
{t, 0, 2 Pi}]

[ア @yuruyurau]

a=(x,y,d=mag(k=4*cos(x/21),e=y/8-20))=&gt;circle((q=3*sin(k*2)+.3/k+sin(y/19)*k*(9+2*sin(e*14-d*3+t*2)))+50*cos(c=d-t)+200,q*sin(c)+d*39-475,k*k&gt;15?2:1)
t=0,draw=$=&gt;{t||createCanvas(w=400,w);background(9).noStroke().fill(w,116);for(t+=PI/240,i=1e4;i--;)a(i,i/235)}#つぶやきProcessing

[Vitaliy Kaurov @superflow]

@rybinfx Yep. The code I posted uses:
https://t.co/O4P9LmlmOA

[Vitaliy Kaurov @superflow]

Three terms to ponder: 𝐛𝐢𝐨𝐦𝐨𝐫𝐩𝐡𝐢𝐬𝐦, 𝐳𝐨𝐨𝐦𝐨𝐫𝐩𝐡𝐢𝐬𝐦, and 𝐚𝐩𝐩𝐚𝐫𝐞𝐧𝐭 𝐚𝐧𝐢𝐦𝐚𝐜𝐲. The perceptual lock is strong: it is impossible not to see an animal. Have you heard of the 1917 book "On Growth and Form" by D’Arcy Thompson (Scottish pioneer of mathematical and theoretical biology)? It made a powerful point: biological shape can be studied as geometry, growth, and transformation. Later, theoretical 𝐦𝐨𝐫𝐩𝐡𝐨𝐬𝐩𝐚𝐜𝐞 made that idea more explicit: vary a few parameters of a geometric model, and you get a space of possible forms. Some are occupied by nature. Some are mathematically possible but biologically absent.

[Vitaliy Kaurov @superflow]

This structure can be made w/ Wolfram 1-liner:
p=Tuples[Range[-2,2],4] . I^{0,1,4/3,7/3}; RelationGraph[Abs[#1-#2]==1&amp;,p,
VertexCoordinates-&gt;ReIm@p]
That's static full graph. RandomSample links, add them 1 by 1, and you get this animation.

[Alvaro Lozano-Robledo @mathandcobb]

Following up on the suggestion from Will Sawin, here is an illustration of the new configurations that disprove Erdos' unit distance conjecture (made with the help of ChatGPT 5.5 Thinking).

[Vitaliy Kaurov @superflow]

ℚuantum VS ℂlassical 𝐆𝐚𝐥𝐭𝐨𝐧 𝐁𝐨𝐚𝐫𝐝
ℂ: one bead, one path, normal distribution from many trials; ℚ: one wave packet, all paths, interference rewrites distribution via probability density of wave function.
#Wolfram code: https://t.co/VwVnmn6vTI

[Vitaliy Kaurov @superflow]

@fermatslibrary One-liner Wolfram code to get all of them:
EntityClass["Polyhedron", "Johnson"]["Graphics3D"]

[Vitaliy Kaurov @superflow]

@fermatslibrary First 𝟏 𝐦𝐢𝐥𝐥𝐢𝐨𝐧 powers of 2 checked with an elegant Wolfram one-liner code. You can add Parallelize to thread of many CUP cores:
Select[2^Range[0,10^6],SubsetQ[{1,2,4,8},IntegerDigits@#]&amp;]

[Vitaliy Kaurov @superflow]

@fermatslibrary The real photo: Regulus Missile Mail at Naval Air Station Mayport, Florida (USA), 8 June 1959

[Vitaliy Kaurov @superflow]

@fermatslibrary But did you know that e^π = (−1)^(-i) (as principal value) and is Gelfond's constant and is a transcendental number. But π^e is a mystery - right now, mathematicians do not even know if π^e is rational. And e^π &gt; π^e :-)

[Vitaliy Kaurov @superflow]

@fermatslibrary Computational proof in Wolfram out 1M integers:
Select[Characters[IntegerName[Range[10^6]]], OrderedQ]
Out[]= {{"f", "o", "r", "t", "y"}}

[Vitaliy Kaurov @superflow]

The cover of The Journal of Physical Chemistry Letters ( FEB 2026).
PAPER: https://t.co/2pXrf8PvLN
CODE: https://t.co/WG7DNSTRuH

[Vitaliy Kaurov @superflow]

Cover of prominent chemistry journal features a stunning structure made with Wolfram Language. When an unbound electron appears in the water a lot of things change around it. It is called a hydrated electron and it is in the center of its environment in the video.

[Vitaliy Kaurov @superflow]

"Most-Viewed People on Wikipedia in 2025", my new article. Novel Social Memory https://t.co/BJxlbOlkIo log ratio of post- to pre- catalyst event median Wikipedia pageviews, measuring how a catalyst event resets collective baseline attention. Comment If you know similar measures.

[Vitaliy Kaurov @superflow]

@AkiyoshiKitaoka Love it! A minimal Wolfram code one-liner: https://t.co/D2V845jcB8
ListPlot[RandomReal[1,{10^4,2}],
Background-&gt;Black,ColorFunction-&gt;
(Blend[Riffle[Table[Red,4],Blue],Mod[#,1]]&amp;)]

[Vitaliy Kaurov @superflow]

⚛️ In 𝐫𝐢𝐯𝐞𝐫 𝐦𝐨𝐝𝐞𝐥 of 𝐛𝐥𝐚𝐜𝐤 𝐡𝐨𝐥𝐞𝐬 space itself flows...
River of space falls into black hole at Newtonian escape velocity, hitting light speed at horizon. Newton particle-grid with Wolfram differential equations gives a qualitative proxy for the visual:

🔴 Wolfram code & article: https://t.co/TukmuKVgeq

ABSTRACT excerpt: "The river model of black holes":
"The river model is mathematically sound, yet simple enough that the basic picture can be understood by non-experts. In the river model, space itself flows like a river through a flat background, while objects move through the river according to the rules of special relativity. In a spherical black hole, the river of space falls into the black hole at the Newtonian escape velocity, hitting the speed of light at the horizon. Inside the horizon, the river flows inward faster than light, carrying everything with it."

[Vitaliy Kaurov @superflow]

How-to "see" 4th dimension in simple steps:
...using 𝐟𝐚𝐦𝐢𝐥𝐢𝐚𝐫 𝐨𝐛𝐣𝐞𝐜𝐭𝐬. 1️⃣ This 4D object = donut;
2️⃣ Cut typical 3D donut across its tube;
3️⃣ The shape of the cut is usual 2D circle;
4️⃣ Generalize by +1: cut 4D donut and get 3D shapes at the cut.

Which is similar to the red tubes in video. Those red tubes are shape of cuts when our familiar 3D space slices 4D donut. And they are similar to our typical 3D donuts!

What's your favorite application of higher dimensions?

TREFOIL KNOT & UNKNOT
Another stunning part in the video is gorgeous 𝐭𝐫𝐞𝐟𝐨𝐢𝐥 𝐤𝐧𝐨𝐭 falling apart into 2 separate rings and then reconnecting again into 𝐮𝐧𝐤𝐧𝐨𝐭 -- starting at t =12 seconds. I highly recommend to pause and slowly scroll through this structure. Its formation is analogical to how you twist 180° a paper band to make a Möbius strip. For details and code see:

🔴 Wolfram code & article: https://t.co/wPeK3AeIIb

APHANTASIA & ABSTRACT THINKING
𝐀𝐩𝐡𝐚𝐧𝐭𝐚𝐬𝐢𝐚 is the inability to visualize in the mind. Some people cannot form images in their thoughts, for example imagining an apple. Surprisingly people with aphantasia have ability to think of higher dimensions through the power of abstraction. Grasping very visual entities even with no ability to visualize. The mind is truly a mystery.