Moritz Schauer

Time for Jenga bridges! It is a classic that you can achieve infinite overhang because the harmonic series 1 + 1/2 + 1/3 + 1/4... diverges (double the numbers in the figure). en.wikipedia.org/wiki/Block-s... #MathSky

The Jenga stacks I built are not very precise and not close to the optimum. But they are better this one where I was stupid for a moment haha

Time for Jenga bridges! It is a classic that you can achieve infinite overhang because the harmonic series 1 + 1/2 + 1/3 + 1/4... diverges (double the numbers in the figure). en.wikipedia.org/wiki/Block-s... #MathSky

The trick is that a cap stone only acts with half of its weight on each pier. Claude and I conjecture that the optimal configuration looks like this.

Time for Jenga bridges! It is a classic that you can achieve infinite overhang because the harmonic series 1 + 1/2 + 1/3 + 1/4... diverges (double the numbers in the figure). en.wikipedia.org/wiki/Block-s... #MathSky

That means for a bridge you can have twice the optimal four-block overhang, (15−4√2)/4 ≈ 2.336 But...

Time for Jenga bridges! It is a classic that you can achieve infinite overhang because the harmonic series 1 + 1/2 + 1/3 + 1/4... diverges (double the numbers in the figure). en.wikipedia.org/wiki/Block-s... #MathSky

Sick child last week - but the scientifically correct spider net kept us entertained.

But the spider web itself is implemented using an ad hoc solver of Hooke's law - the solver iteratively moves each free node in the direction of its net force. That is why adding a radius visibly shifts the hub and deforms the frame rather than simply drawing a new straight segment.

Sick child last week - but the scientifically correct spider net kept us entertained.

And for all us children that like to build correct spiderwebs in their living room, I thought I try Claude with manim to animate and narrate this. Amazing, honestly m.youtube.com/watch?v=qF87...

#MathSky Trying to demystify Caratheodory's measure extension theorem: mschauer.github.io/share/friend...

Furstenberg gave a fascinating proof that extracts the infinitude of the primes from seemingly innocuous manipulations of clopen sets. Here is a retake: The σ-algebra 𝓕ₚ = σ(n ↦ n mod p) encodes exactly the information contained in knowing n mod p. That is natural, σ-algebras model information.

Here is Furstenberg's topological version: en.wikipedia.org/wiki/Fursten...

The σ-algebra viewpoint also gives a nice form of the Chinese remainder theorem: for pairwise coprime n₁, …, nₖ it holds σ(x mod n₁, …, x mod nₖ) = σ(x mod n₁⋯nₖ). Knowing the residues separately is exactly the same information as knowing the residue modulo the product.

This is fun! 𝓕(m) ∨ 𝓕(n) = 𝓕(lcm(m,n)), 𝓕(m) ∩ 𝓕(n) = 𝓕(gcd(m,n)).

Furstenberg gave a fascinating proof that extracts the infinitude of the primes from seemingly innocuous manipulations of clopen sets. Here is a retake: The σ-algebra 𝓕ₚ = σ(n ↦ n mod p) encodes exactly the information contained in knowing n mod p. That is natural, σ-algebras model information.

Since every integer except −1 and 1 has a prime divisor, ℤ ∖ {−1,1} = ⋃ₚ pℤ, so {−1,1} ∈ 𝓕. Each 𝓕ₚ contains the translations of its elements, hence so does 𝓕. So {−1+k,1+k} ∈ 𝓕 for every k ∈ ℤ. These sets are all distinct, so 𝓕 is infinite...

Furstenberg gave a fascinating proof that extracts the infinitude of the primes from seemingly innocuous manipulations of clopen sets. Here is a retake: The σ-algebra 𝓕ₚ = σ(n ↦ n mod p) encodes exactly the information contained in knowing n mod p. That is natural, σ-algebras model information.

Probability theory causes a certain mathematical inflation: random variables are functions, events are sets, and probabilities are countably additive set functions of total mass one defined on σ-algebras.

The redeeming feature is that this price is paid only once.

Gaurav Arya’s presentation at PLDI'26 on "Gradient Estimation as Probabilistic Inference" bringing sound, compositional and automatic gradient estimation for expectations in probabilistic programs. Great job, Gaurav!! www.youtube.com/watch?v=0aWz...

Causal inference is often hidden in plain sight. In a randomized clinical trial, the setup is such that interventional and conditional distributions coincide. That is E(X | do(T = t)) = E(X | T = t).