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Animated lectures on numerical methods & applied math — clear visuals, rigorous explanations, verified computations.

28/08/2026

You've seen this spiral on posters, book covers, and a thousand design
blogs. Almost none of them mention that it's a proof.
Take the Fibonacci numbers — 1, 1, 2, 3, 5, 8, 13 — and square each one. That's 1, 1, 4, 9, 25, 64, 169, adding to 273.

Now draw each as an actual square and lay them against each other. They tile
a 13 by 21 rectangle exactly. No gaps, no overlaps. And 13 × 21 = 273.

It can't fail either. Every new square is exactly as long as the flat side
of the block you already have, so the two sides of the rectangle keep adding
— which is the Fibonacci rule itself. 8 + 13 = 21.

Sweep a quarter circle through each square and you get the spiral.

From Roger Nelsen's Proofs Without Words, p. 83, after Alfred Brousseau.

27/08/2026

4 + 5 + 6 + 7 + 8 + 9 + 10 = 49.

Seven consecutive numbers, and the total is seven squared. Worth a second
look.

Draw each number as a row of tiles and you get a lopsided staircase. Now
even it out — take the overhang off the long rows and use it to fill in the
short ones. The 10 gives three to the 4, the 9 gives two to the 5, the 8
gives one to the 6, and the 7 in the middle never moves.

Every row becomes seven long, and there are seven rows.

The rule underneath is a useful one: any odd-length run of consecutive
numbers adds up to the count times the middle term. Try 9 through 25 —
seventeen numbers, middle term 17, total 289.

25/08/2026

Count up to six and back down again: 1+2+3+4+5+6+5+4+3+2+1.
That's 36. A perfect square. And it works whatever number you count to.
The reason is hiding in the diagonals. Take a 6×6 block of tiles and slice it corner to corner. The diagonal stripes are 1, 2, 3, 4, 5, 6, 5, 4, 3, 2, 1 tiles long. They grow to the middle and shrink back, because that's
simply what the diagonals of a square do.
Cut along the longest one and you're left with two staircases, 21 tiles and 15 tiles. Two consecutive triangular numbers making a square.

24/08/2026

Add the odd numbers in order: 1, then 1+3, then 1+3+5, then 1+3+5+7.

You get 1, 4, 9, 16 — perfect squares, every single time.

It isn't luck. Each odd number is exactly the L-shaped shell that wraps
around the square you already have. To grow a 5×5 square into a 6×6, you
add a column of five, a row of five, and the single corner where they meet.
Five plus five plus one is eleven — and two equal piles plus a leftover is
always an odd number.

Nicomachus of Gerasa worked this out around AD 100.

From Roger Nelsen's Proofs Without Words, p. 71.

22/08/2026

Add every number from 1 to 10 — without doing a single addition.

Stack them as rows of tiles and you get a staircase. Take a second copy,
turn it upside down, and the two halves lock into a perfect rectangle:
10 rows, 11 columns, 110 tiles. So the staircase you started with is
exactly half of that. 55.

The same picture works for any number you like, which is where the
formula n(n+1)/2 actually comes from.

19/08/2026

A circle is secretly a triangle.

Cut it into rings instead of slices. Each ring at radius s has length 2πs.
Unroll them, stack them longest at the bottom, and you get a triangle with base
2πr and height r — so the area is ½(2πr)(r) = πr².

The height is r because you cut from the centre out to the rim.

Each ring is one term of A = ∫₀ʳ 2πs ds. The picture is the integral.

14/08/2026

This pizza turns into a rectangle. Two-thousand-two-hundred-year-old math, still works.

11/08/2026

Normally more measurements mean a better result. Not always. Fit a curve through 5 evenly spaced points and it's already rough. Through 17, and the ends of the curve plunge wildly downward — on a function that never even goes negative.

The middle got better every time. It's the edges that break.

10/08/2026

A temperature sensor doesn't actually measure temperature. It produces a voltage, and the factory has to translate. They measure it at three known temperatures, fit a curve through those three points, and that curve handles everything in between.

Simple idea, and it's inside almost every device you own.

05/08/2026

Given three points on a curve, you can work out the height anywhere in between — without ever figuring out what the curve actually is. Just blend the numbers in pairs, then blend the results.

Sailors used this method to find their longitude before GPS existed.

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