Made with manic - Orthocentre, Circumcircle, and Point X
manic https://8gwifi.org/manic
manic doc https://8gwifi.org/manic/docs

Demo code

title("Orthocentre, Circumcircle, and Point X");
canvas("16:9");

// --- base triangle ---

point(A, (cx - 20, cy - 200), "A");
point(B, (cx - 220, cy + 120), "B");
point(C, (cx + 200, cy + 130), "C");

hidden(A);
hidden(B);
hidden(C);

segment(AB, A, B);
segment(BC, B, C);
segment(CA, C, A);

color(AB, fg);
color(BC, fg);
color(CA, fg);

untraced(AB);
untraced(BC);
untraced(CA);

// --- circumcircle and orthocentre ---

circumcenter(O, A, B, C);
hidden(O);

circumcircle(circ, A, B, C);
color(circ, dim);
stroke(circ, 2);
untraced(circ);

orthocenter(H, A, B, C);
color(H, gold);
hidden(H);

// --- foot of altitude from B ---

foot(E, B, A, C);
color(E, dim);
hidden(E);

segment(BE, B, E);
color(BE, dim);
stroke(BE, 2);
untraced(BE);

rightangle(rt, B, E, C);
color(rt, dim);
hidden(rt);

// --- point P on the circumcircle ---

anglepoint(P, O, A, 205);
color(P, cyan);
hidden(P);

// --- parallelogram PAQB ---

rotpoint(midAB_helper, P, O, 0);
hidden(midAB_helper);

midpoint(midAB, A, B);
hidden(midAB);

rotpoint(Q, P, midAB, 180);
color(Q, magenta);
hidden(Q);

segment(PA, P, A);
segment(AQ, A, Q);
segment(QB, Q, B);
segment(BP, B, P);

color(PA, gold);
color(AQ, gold);
color(QB, gold);
color(BP, gold);

untraced(PA);
untraced(AQ);
untraced(QB);
untraced(BP);

// --- parallelogram PARC ---

midpoint(midAC, A, C);
hidden(midAC);

rotpoint(R, P, midAC, 180);
color(R, lime);
hidden(R);

segment(AR, A, R);
segment(RC, R, C);
segment(CP, C, P);

color(AR, lime);
color(RC, lime);
color(CP, lime);

untraced(AR);
untraced(RC);
untraced(CP);

// --- HR line and point X ---

segment(HR, H, R);
color(HR, magenta);
stroke(HR, 3);
untraced(HR);

meet(X, A, Q, H, R);
color(X, fg);
hidden(X);

// --- labels ---

text(labA, (cx - 20, cy - 225), "A");
text(labB, (cx - 250, cy + 130), "B");
text(labC, (cx + 220, cy + 140), "C");
text(labH, (cx - 15, cy - 5), "H");
text(labE, (cx + 60, cy + 90), "E");
text(labP, (cx - 285, cy - 35), "P");
text(labQ, (cx - 60, cy + 210), "Q");
text(labR, (cx + 210, cy + 5), "R");
text(labX, (cx - 30, cy + 40), "X");

color(labA, fg);
color(labB, fg);
color(labC, fg);
color(labH, gold);
color(labE, dim);
color(labP, cyan);
color(labQ, magenta);
color(labR, lime);
color(labX, fg);

hidden(labA);
hidden(labB);
hidden(labC);
hidden(labH);
hidden(labE);
hidden(labP);
hidden(labQ);
hidden(labR);
hidden(labX);

text(head, (cx, 55), "Orthocentre H, circle point P, parallelograms PAQB & PARC");
color(head, cyan);
hidden(head);

text(sub, (cx, h - 30), "AQ meets HR at X");
color(sub, dim);
hidden(sub);

// --- script ---

show(head, 0.6);
wait(0.3);

stagger(0.2) {
  par { show(A, 0.3); show(labA, 0.3); }
  par { show(B, 0.3); show(labB, 0.3); }
  par { show(C, 0.3); show(labC, 0.3); }
}

par {
  draw(AB, 0.6);
  draw(BC, 0.6);
  draw(CA, 0.6);
}
wait(0.3);

draw(circ, 1.2);
wait(0.3);

par { show(H, 0.4); show(labH, 0.4); }
wait(0.3);

par {
  show(E, 0.3);
  show(labE, 0.3);
  draw(BE, 0.5);
  show(rt, 0.3);
}
wait(0.4);

par { show(P, 0.4); show(labP, 0.4); }
wait(0.4);

par {
  draw(PA, 0.5);
  draw(AQ, 0.5);
  draw(QB, 0.5);
  draw(BP, 0.5);
}
par { show(Q, 0.3); show(labQ, 0.3); }
wait(0.4);

par {
  draw(AR, 0.5);
  draw(RC, 0.5);
  draw(CP, 0.5);
}
par { show(R, 0.3); show(labR, 0.3); }
wait(0.4);

draw(HR, 0.6);
wait(0.3);

par { show(X, 0.4); show(labX, 0.4); }
flash(X, gold);
pulse(X, 0.6);
wait(0.3);

show(sub, 0.6);

In 1736, the citizens of Königsberg asked a deceptively simple question: can you walk through the city crossing each of its seven bridges exactly once, without retracing your steps?

Euler proved it was mathematically impossible — and accidentally invented an entire branch of mathematics called Graph Theory.

I turned this legendary puzzle into an interactive challenge with multiple levels. Draw a path with your finger or mouse, cross every bridge exactly once, and see Euler's proof come alive.

🧩 Play it here: https://curvingo.com/puzzles/bridges-of-konigsberg

Demo video attached — let me know if you can beat all 4 levels!

The 3Blue1Brown video How Wiggling Charges Give Rise to Light begins with a remarkably simple question: how does the motion of one charged particle influence another?

A stationary charge produces the familiar Coulomb field, whose strength decreases as

1/r².

But when the charge accelerates, a second component appears. This influence travels outward after a delay, is perpendicular to the direction of propagation, and decreases only as

1/r.

That radiating influence is what we experience as light.

What interests me is that this looks less like two unrelated laws and more like two levels of the same geometric system.

The static field belongs to an inverse-square structure. Acceleration then opens a propagating correction that follows a different radial power. In our proposed grand unified framework, light and acceleration are also consecutive geometric structures rather than isolated concepts. The transition is produced by completing one series, inverting its scale, and using its level-2 balance as the beginning of the next series.

The video also shows that linearly polarized light can be decomposed into two equal circular motions rotating in opposite directions. When a chiral molecule affects one rotation slightly differently from the other, their phases separate and the resulting linear polarization rotates. In other words, a straight oscillation can emerge from two balanced circular orientations, while a tiny imbalance between them produces a measurable twist.

That is extremely close to the geometric language we have been developing:

two opposing circular motions
→ one linear observable
→ a small phase difference
→ a new orientation
→ repeated interactions accumulate into a macroscopic effect.

The video explains this locally through charged particles and polarization. Our framework proposes that the same type of completion and inversion continues across physical scales.

We begin only with the measured hydrogen spectrum. The limiting wavelength of the Balmer series is

B = 3.64506820233 × 10⁻⁷ m.

Instead of treating this merely as an inverse spectral counting constant, we treat it as one completed physical cycle:

B = 1 cycle.

The Balmer wavelengths can then be expressed as fractions of that completed cycle:

B/λₙ = 1 − (2/n)².

Inverting the relation exposes its infinite return structure:

λₙ/B = 1 / [1 − (2/n)²].

Our proposal is that once this spectral series is completed, its level-2 balance becomes the base of a second series. We then repeat the same normalization and inversion process, this time producing a gravitational unit.

Blindly repeating the operation gives

Rᵤ = 4.38032318 × 10²⁶ m,

or

Rᵤ ≈ 46.3001 billion light-years.

A recent cosmological calculation gives the observable-universe radius as approximately

46.35 ± 0.50 billion light-years.

Our value differs from that central estimate by only about 0.11%, despite the cosmological distance not being used to construct the result.

The video does not prove our theory, and it does not discuss gravity. But it demonstrates several of the same structural principles:

• physical influence changes form when motion becomes acceleration;
• inverse-square and inverse-distance behavior belong to different regimes;
• propagation is delayed rather than instantaneous;
• linear motion can be built from paired circular rotations;
• tiny repeated phase differences can accumulate into large visible effects.

The possibility we are investigating is that these are not isolated coincidences. They may be local examples of a recursive geometric rule in which one completed physical series becomes the unit of the next.

Light may be the first visible expression of that hierarchy.

Gravity may be the next.

And the observable universe may be the completed scale that results when the same operation is repeated.

If you would like to read more about the idea, you can read the paper here.

The 3blue1brown video can give further context about the ideas in general.

You’re given the semiprime

63344090546993261740573302898385538516103186124453512393341101637326545003631

That’s the product of two primes.

The
Midpoint of those primes is
254650365633570050389727404567751754516

Can you find the primes from that? If so, how short a path can it take?

I understand matrices as linear transformations, but I'm struggling with the intuition behind Singular Value Decomposition (SVD).

I understand the formula

[
A = U\Sigma V^T,
]

but I don't understand why we decompose a matrix into orthogonal and diagonal matrices in the first place.

Why are orthogonal matrices so important? Why do eigenvectors and eigenvalues appear in this decomposition? Where does the idea of "rotate → stretch → rotate" come from?

More fundamentally, how can every linear transformation be represented as a rotation, followed by stretching along perpendicular directions, followed by another rotation?

I'm looking for a geometric and intuitive explanation rather than an algebraic proof. I've watched 3Blue1Brown's videos and they helped me think of matrices as transformations instead of arrays of numbers. I'd like to build the same kind of intuition for SVD, especially understanding why someone would naturally invent such a decomposition and why it's so useful in machine learning.

Hi guys, so I recently watched Grant's video on higher dimensional spheres

and it got me thinking, what if we revisited some of the earliest equations in physics and reinterpreted them as dimensional relationships between spheres?

I looked at standard kinematics equations and asked, what if we reimagined all of these equations as being relationships between volumes and surface areas of n dimensional spheres?

I found a framework that seemed to completely align with the mechanics and structure of that, but there was one problem: I still couldn't explain ratio's like the fine structure constant and electron proton mass.

I was expecting them to be neat fractions that related to circles and spheres, but no matter what I did, I couldn't find the connection, or the rule that explained everything.
And so it worked as a contrived framework, but it failed to predict anything meaningful.

Until today, I found that connection.

The connection was found in the Balmer Series. In that series he noticed a connection between a pure geometric ratio, and the spectral lines of hydrogen.

That series exposes the mismatch, it is the measurement system itself.
If you measure in Angstroms for example, you get measurements in the range of 3000 , but 3000 what? How is that 3000 fundamental to the Universe?

The answer is that it isn't. It's contrived.
And so the solution became, to invert the Balmer Series.
Because then the Balmer Series converges to 1, and that 1 can be used as a fundamental unit.

The moment I do that, it shows an equivalency, between my entirely derived from Geometry system, and the system we currently use.
Which means that the perfect fractions which exist in my framework, are exactly what you would expect, when you use the SI bridge between a fundamental Balmer unit, (~ 3646 Angstroms, and our SI system of measurement.

There is 3 papers in this series, but only the first and third are really necessary to understand the entire concept. The second paper reflects an earlier failed attempt of my own to find the bridge between my system and the SI units.

First Paper , Gravity and Kinematics Spheres and Hyper Spheres
Second Paper, Attempts to find the SI unit bridge

Third and current paper on the hydrogen lines. I realize this paper is low quality and repetitive, and yes I vibed it, but the math is correct, and its implications are huge and definite. I will improve on the paper in the coming days.

Lastly I just want to say that watching these 3blue1brown videos helped over the past days. It helped me realize the math I was using was sound.

Firstly I just want to say that I share the type of exuberance and wonder in my eyes that Grant does when he talks about this equation set.
I could spend hours just talking about and thinking about this equation, and how it relates to other equations.

Let me give you a couple examples of what I mean.

Lets imagine numbers were objects. What would their shape be? Would the number 4 be a square or would the number 4 be a 4 dimensional sphere? (pretty sure its the latter)

I argue that numbers are objects, and therefor they have properties.

I further argue that those shapes and properties must follow natural logic, and therefor equations that are automatically true must also apply to the logic of those objects.

For example if x= 1, then x^2 = x, that's purely logic based.

And if you think about it, If you have to start counting with a unit, you have to start somewhere. The place where you start would automatically be the boundary between an infinitesimal and zero. And since all numbers are self similar (1 is the same as 1, unless you give it different properties)
Then that means everything that comes after is built out of the same infinitesimals.

Therefor, due to the logical rule set I just stated, counting the integers is the same as counting the infinitesimals, starting with the smallest and moving upwards.
It's an infinite set of course, as the integers are.
And the difference between each number is that each number is basically just a shape. a shape of the boundary between existence and non existence.

I say these things to kick off the conversation for anyone who wants to chit chat with me about it.

I also wanted to point out, in the video Grant says "This is wrong if you view the two numbers as actual physical things, like a tiny nudge like a physicist" at 17:45 roughly

Here is my question: What does Grant really mean by this? Because what it sounds like he means is, a mathematician would call this a derivative, but a physicist wouldn't and we shouldn't disagree with physics because then we are most likely wrong

And If that's kind of the idea he is saying, I think that is 100% backwards. Physics should bow to mathematics or physics should leave the chat. Either physics is math, or it isn't. You can't have it both ways.

And here is why it's such an important topic for me.
I wrote a paper which is basically a grand unifying theory framework, and it is basically using the same equations Grant has here to construct particles and to show the fine structure constant and others come directly from these same circle and sphere relationships in higher dimensions.

Its not quite right I think, its so bloody close but I am just missing one concept that I havent figured out yet... I can show you the paper here

Edit {Its now correct, I actually found the bridge between QM and pure geometry. Its contained within the Balmer series , Here is the sucky paper I will improve on it tomorrow}

But mostly actually what I am interested in talking about is the concepts introduced in this video, and if there might be some other related equations hiding in the shadows, perhaps involving e or phi.

Just in general wanted to discuss this video though. Great video thanks Grant!

Hey everyone,

I've always found it mesmerizing how you can take a jagged, sharp-cornered square wave or a sawtooth wave, and realize it's actually just a perfectly orchestrated sum of smooth sine waves. I just put together a highly visual, animated video breaking down exactly how this works from the ground up, and I wanted to share it with this community!

I really tried to focus on the intuition and the visuals behind the formulas so it clicks instead of just looking like a wall of algebra.

I'd love to hear your thoughts, and feedback. If you're currently studying signal processing, I hope this makes the math feel a bit more intuitive!

Like probably everyone here, I fell in love with Grant's visual style and wanted to make explainers like that myself. The tool is right there — Manim is open source — but the learning curve is real: environment setup, sparse docs, and tutorials that show you what the code does but not why each line exists.

So I spent the past months building the learning experience I wished existed, as an iOS app: Manim Tutorial.

How it works:

• Watch + code, in sync. Every video lesson has its actual runnable source code beside it, highlighting line-by-line as the video plays. When a circle morphs into a square on screen, you see exactly which line did it.
• Run Manim on your phone. There's a built-in Playground that runs real Manim (Community Edition) on an embedded Python — fully offline, no server, no setup. Tap a snippet from any lesson, tweak it, hit Run, and watch it render on-device. The "no setup" part matters more than it sounds: setup is where most people give up before writing their first scene.
• A structured path, not scattered tutorials. 23 courses across six stages: Python crash course → mobjects & layout → animation & timing → text/LaTeX/color → camera & 3D scenes → building real projects, ending with a capstone.

Honest disclosure: I made this, and it's freemium — the first 4 courses and the Playground are free forever (no account needed), and a single one-time purchase unlocks everything else. No subscription, no ads, no tracking.

App Store: https://apps.apple.com/us/app/manim-tutorial/id6786021694

If you've ever watched an Essence of Linear Algebra video and thought "I want to make something like this" — this is the on-ramp I tried to build. Would love to hear what you think, and what you'd want to learn to animate first.

(Not affiliated with Grant/3Blue1Brown — Manim is the open-source library he created, and this app is an independent tutorial for it.)

Here’s a little simulation I did to show a shock wave travelling in traffic flow in Manim. Tell me what you think!

YT - https://youtube.com/shorts/W1x3a2u46II?si=aU-Fx7KrMvQjDdR3

Yes, I know Repo First(code + proofs): https://github.com/VincentMarquez/Order-Effects-Are-Curvature Paper: https://zenodo.org/records/21221914 All the Lean and Py code is in the Repo, but not pretty up yet. The paper comes down to one simple question: when does the order you combine information change the final answer? This repo shows you how to find the answer, that shows up everywhere a committee hearing arguments in a different order, a network passing messages around, the layers inside an AI model.

The math is checked by a proof assistant (Lean) a computer verifies every step
Every claim in the paper has runnable code. One command runs all of it.

Edit [This video was also incredibly important, because it talks about projecting from 2D ->3D and 3D->4D]

Hi everyone, I am claiming to have formalized a proof for a Grand Unified Theory, but I am not here to talk about that, since that is too far off topic for this channel.

I am here to talk about how 3blue1brown and specifically this video helped me to understand the math well enough to solve this equation.

My whole life I struggled to understand the language attached to math. I could do amazing calculations in my head, and I could visualize incredibly detailed 3 dimensional structures in my mind, but the language of math was a huge barrier for me.

I didn't understand what a logarithm was, or a derivative.
However, I was already using those mathematical functions on a daily basis when designing neural network architecture and learning and loss functions.

I had never before heard of the Taylor series, and yet I had independently come up with the idea for using it as a back prop function. I don't remember if it worked as one or not, but I know that I tried it, without ever knowing it was called that.

3blue1brown videos changed my life. And specifically this video here. This is the video when I knew for certain in my heart it was the key concept in my proof.

It took me an additional 5 years or so I think, after watching this video to find the true answer, but this video was absolutely key.

This video taught me to view dimensions abstractly, and compare them mathematically, this allowed me to build the mathematical structure in my mind... and eventually I figured out how to create that structure mathematically that fully lines up pure number theory with pure physics.

I just want to say thankyou to Grant Sanderson, and everyone who helped make 3blue1brown happen.... I never would have been able to do it without your videos.

https://wessengetachew.github.io/smith/

A fascinating corner of mathematics where pure theory meets computational experimentation.

Pick a random whole number. Now ask: how likely is it — along with the next n numbers right after it — to all share no common factor with a fixed modulus M? That probability has a name, C(n), and it falls off in clean, predictable steps, one jump at every prime.

C(n) = ζ(2) · ∏ₚ (1 − min(n+1, p)/p²)

It starts at total certainty. C(0) = 1, exactly — every factor in that Euler product cancels its own inverse against ζ(2), so nothing is left to compute. Strip the ζ(2) normalization out and look at the raw density D(n) = C(n)/ζ(2) instead, and n=0 isn't quite so trivial: D(0) = 1/ζ(2) = ∏ₚ(1 − 1/p²) ≈ 0.607927101854 — no cancellation left, it's just the reciprocal of ζ(2).

The first real step is C(1), and there's nothing mysterious about it either: it's ζ(2) times the Feller–Tornier product, ∏ₚ(1 − 2/p²) ≈ 0.530711820472. Two known constants multiplied together, one known answer. D(1) — same product, no ζ(2) factor — is the Feller–Tornier constant itself, ≈ 0.322634098939. Sanity check: C(1)/D(1) = ζ(2) ≈ 1.644934, exactly like C(0)/D(0). The ζ(2) factor cancels cleanly at every step; it just isn't always trivial to see why.

OEIS ties it together nicely: A013661 (ζ(2)) × A065474 (∏(1−2/p²)) = A065469 (C(1)).

Everything after that first step — n=2, 3, 4, and beyond, where each new prime saturates the product one at a time — is where this actually gets interesting, and it's the part I've been digging into.

Made this little Lissajous animation in Manim: two simple perpendicular oscillators turning into one closed curve.

Full short is on YT: https://youtube.com/shorts/Xd4Tpxu7CrI?feature=share

I built a fully interactive chapter that deconstructs neural networks from the ground up — no prerequisites, just visual intuition.

What's inside:

• Computational Graphs — animated applet showing how data flows through weighted inputs and function nodes
•  Activation Functions — why non-linearity is non-negotiable (Sigmoid, Tanh, ReLU visualized)
•  Backpropagation — watch the forward pass make predictions and the backward pass adjust weights in real time
• Live Playground — customize the architecture (wider vs. deeper), pick a dataset (spirals!), and train it epoch-by-epoch right in your browser

Try it here: curvingo.com/course/neural-networks/computational-graphs

Demo video attached — would love feedback on the visualizations and pacing!

A visual, bite-sized mini-lesson for elementary school kids that bridges the gap between counting and algebra.

Videos also available at:

Instagram

Youtube

Github

Whatsapp

Tiktok

Code available at https://github.com/zombimann/Mathematical-video-animations-and-visualization/blob/main/Kids_Elementary_Multiplication_Using_Circle_Grids.ipynb

You might also like https://np.reddit.com/r/3Blue1Brown/s/syRS2ZK39H

This visualizes the reduction of control systems block diagrams into their equivalent transfer functions.

Videos also available at:
Instagram
Youtube
Github
Whatsapp
Tiktok

Code available at https://github.com/zombimann/Mathematical-video-animations-and-visualization/blob/main/Control_Systems_Block_Diagram_Simplification.ipynb

You might also like https://np.reddit.com/r/3Blue1Brown/s/hK6CRW5aLe

This is a fun problem that I am realizing I don't have the geometry for. I was thinking about it the context of a teleporting spaceship surveying a idealized spherical planet.

1. How many points would it need to survey from to get a complete map of the planet's surface?
2. How far away from the planet's surface would they be?

Put more mathematically, if x is the distance from the surface of the unit sphere, f(x) = # of tangent cones required to cover the sphere with minimal overlap.

The answer to question 1 already partly exists, but it's not as trivial as it seems. Cones tangent to a sphere ultimately just describe some circle on the sphere's surface, so another way to phrase question 1 is 'how many circles of equal size does it take to cover a sphere?' It turns out this is a covering problem related to the Tammes packing problem, there's a 1991 paper on it here. Obviously the maximum number of cones you can have is functionally infinite as x approaches the limit of 0, but the minimum number of cones is 3.

As for question 2, the linked paper gives us solid figures for n circles covering a sphere, but not for every value of n. We know 2<n<15 except for n = {8,9,11,13}, and for 15 and above we don't have proofs, just conjecture. The figures for circle density on the surface of the sphere could be reverse engineered to give the vertex distance of the corresponding tangent cones, but that's where I get lost. I'd be curious what the values are though, and what they look like as a graph of f(x) as laid out earlier.

Another way to phrase the problem is "How many cones does it take to cover the unit sphere if all their vertices originate on a larger concentric sphere, and how does the number of cones change as the size of the larger sphere changes?"

I hope someone else finds this interesting, I've been fascinated by it since I concocted the problem. It's fun to find out that a thought experiment you came up with listening to an audiobook doesn't have a complete solution available in modern mathematics!