Viktor Futó · Stay human

@optimista.bsky.social

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Spatially written math & proofs with LaTeX - brainec.com Alumni: University of Coimbra · Charles University in Prague

A screen recording of a spatial mathematical story visualizing Corollary 12.3 from braid-Floer homology. The canvas shows a white background with text nodes arranged in a clean diagram. At the top is the title “Corollary 12.3” with a note that positive braids realize, up to shifts, all braid Floer homologies. On the left, under “Shift Theorem,” appear graded isomorphisms such as HB*((X rel Y)·Δ²) ≅ HB*−2n(X rel Y), extended through Δ²g to HB*(X rel Y) ≅ HB*+2gn((X rel Y)·Δ²g). On the right, under “Garside’s Theorem,” the diagram states that any braid β admits a unique form β = P·Δ⁻²g, with P positive and Δ⁻²g representing negative full twists. Arrows lead to the final conclusion HB*(X rel Y) ≅ HB*+2gn(P), with P = [X₊ rel Y₊]. During the video, the full diagram hides and then reappears through progressive reveal as nodes fade in sequentially.
Me standing in front of a concrete wall with a sign that reads “Departamento de Matemáticas.” The sign lists the building levels and offices. I am wearing a leather jacket and scarf, smiling slightly, with sunlight illuminating my face.
A spatial story presents Merten’s Second Theorem. The canvas defines x as a real number with x ≥ 2 and p as a prime. It then introduces the prime harmonic sum up to x, showing that the sum of 1/p for p ≤ x equals log(log x) plus the Meissel–Mertens constant M, with an additional small correction term O(1/log x). The constant M is defined below, with its approximate value 0.26149721284. The story ends by displaying an equivalent limit form: as x approaches infinity, the difference between the prime harmonic sum and log(log x) tends to M. The video shows the complete spatial story, a progressive reveal, a timelapse of its creation, a slower reveal, and ends with the text “write spatially” and “brainec.com”.
A spatial story explains Euler’s Totient Function. The canvas begins by defining a natural number n and the function φ from ℕ to ℕ. It introduces φ(n) as the count of positive integers up to n that are relatively prime to n, written as the set of k in ℕ where gcd(k, n)=1. A note clarifies that “relatively prime” means “coprime,” i.e., gcd(a, b)=1. The story includes an example showing φ(9)=6, with the integers 1, 2, 4, 5, 7, and 8 highlighted. The video displays the full spatial story, followed by a progressive reveal, a timelapse of its creation, a slower reveal, and ends with the text “write spatially” and “brainec.com”.
A spatial story presents Merten’s Third Theorem. The canvas begins by defining x as a real number with x ≥ 2 and p as a prime number. It then introduces the product over all primes up to x of (1 − 1/p), showing that this product equals e⁻ᵞ divided by ln(x), multiplied by (1 + o(1)). The constant γ is identified as the Euler–Mascheroni constant, approximately 0.57721566490... The story concludes with an equivalent limit form: the limit as x approaches infinity of ln(x) times the same prime product equals e⁻ᵞ. The video displays the full spatial story, followed by a progressive reveal, a timelapse of its creation, a slower reveal, and ends with the text “write spatially” and “brainec.com”.
A spatial story illustrates the Mercator Series, the power series expansion of the natural logarithm. The canvas begins with the condition x ∈ ℝ and |x| < 1, then reveals the main formula ln(1 + x) = ∑ from n=1 to ∞ of ((-1)ⁿ⁺¹ xⁿ / n). The expression unfolds term by term into its explicit form: ln(1 + x) = x − x²/2 + x³/3 − x⁴/4 + … . The video displays the full spatial story, followed by a progressive reveal, a timelapse of its creation, a slower reveal, and ends with the text “write spatially” and “brainec.com”.
A spatial story presents the Euler Product for Dirichlet Series. The canvas begins by defining a multiplicative function f from the natural numbers to the reals and a real parameter s. It then introduces the Dirichlet series, the infinite sum of f(n) divided by nˢ, assumed to converge absolutely. The story reveals the main result: this sum equals the infinite product over all primes p of the inner sums ∑ f(pᵏ)/pᵏˢ. The visual layout connects the global series to its prime factorization structure, illustrating how multiplicativity leads to the Euler product identity. The video shows the full spatial story, followed by a progressive reveal, a timelapse of its creation, a slower reveal, and ends with the text “write spatially” and “brainec.com”.
A spatial story presents Abel’s Summation Formula. The canvas begins by defining a function f on the interval [1, N] that is continuously differentiable and a sequence of complex numbers a₁, …, aₙ. It introduces A(x) as the cumulative sum of aₙ for n ≤ x. Then it reveals the main formula: the sum from n=1 to N of aₙf(n) equals A(N)f(N) minus the integral from 1 to N of A(x)f′(x)dx. The lower half sketches the proof step by step, showing how the discrete sum transforms into an integral using the fundamental theorem of calculus. The story visually connects algebraic and analytic expressions, ending with the boxed final result. The video shows the complete story, a progressive reveal, a timelapse of its creation, a slower reveal, and ends with “write spatially” and “brainec.com”.
A spatial story introduces the concept of a multiplicative indicator function. The canvas first defines a subset S of a set X and the indicator function f(x), which equals 1 if x belongs to S and 0 otherwise. It then transitions to the arithmetic setting, where X is the set of natural numbers and multiplicativity is defined only for arithmetic functions. Two coprime integers a and b are chosen, and the story shows that f is multiplicative if and only if f(1)=1 and f(ab)=f(a)f(b). Blue annotations clarify that the discussion applies to functions on the natural numbers. The video follows the usual format: the complete story, a quick progressive reveal, a timelapse of its creation, a slower reveal, and ends with the text “write spatially” and “brainec.com”.
A spatial story introduces the concept of a modular inverse. The canvas builds gradually: integers a and x are defined, with gcd(a, x)=1. Then it shows that there exists an integer b such that ab ≡ 1 (mod x), meaning ab leaves remainder 1 when divided by x. The story contrasts this with the ordinary non-modular multiplicative inverse, using the example of 5 and 1/5 to highlight the modular analogue. The video first shows the complete spatial story, then its progressive reveal, a timelapse of its creation, a slower reveal again, ending with the text “write spatially” and “brainec.com”.
A spatial story reveals Dirichlet’s Theorem on Arithmetic Progressions. The canvas builds step by step: two positive integers a and b are chosen, and if gcd(a,b)=1 (they are coprime), then there exist infinitely many primes p in the progression p ≡ a (mod b). An example is shown with a=2, b=5: since gcd(2,5)=1, there are infinitely many primes congruent to 2 modulo 5. The sequence appears with examples 2, 7, 17, 37, 47, 67, 97, 107, … demonstrating primes that leave remainder 2 when divided by 5. Video shows in sequence not only the revelation of this spatial story, but also the process of its writing in quick timelapse.
A spatial story reveals Density Lemma 4.4, a key ingredient for proving the density-one Davis–Lelièvre conjecture and the process of writing it in a timelapse. The canvas builds step by step: first the setup with positive integers A, B, C and gcd(A,B)=1, then the linear form l(x,y)=Axy+Bx+Cy whose values are highlighted. The story shows that l represents a density-one subset of positive integers, with the limit formula expressing that the proportion of numbers representable by l approaches 1. Finally, the video refines the statement, showing the precise asymptotic formula with a Big-O error term involving log and Euler’s totient function φ(A). Arrows connect assumptions to conclusions, illustrating the logical flow.
A spatial story reveals the density-one version of the Davis–Lelièvre conjecture. The canvas shows the title and explains that almost all positive even integers occur. A central formula appears as a limit: the ratio of the number of even n ≤ X that occur to X/2 tends to 1. Notes clarify that “occurs” is to be found in the Davis–Lelièvre conjecture post and that X/2 is used because we only consider positive integers. The story refines the result: the number of even n up to X that do not occur is much smaller than X divided by (log X)^(1/40), using Vinogradov notation. Finally, the equivalent expression is shown in Big-O notation: the ratio equals 1 plus a small error term of order 1 over (log X)^(1/40).
The Davis–Lelièvre Conjecture states that every positive even integer except 2, 12, 14, and 18 occurs as the combinatorial period length of an asymmetric closed billiard trajectory on the regular pentagon. The spatial story shows two pentagons with billiard paths: on the left a symmetric closed trajectory with 5 bounces, on the right an asymmetric closed trajectory with 6 bounces. Arrows trace the ball’s movement, returning to the same position and direction, illustrating how “occurs” means the existence of such a trajectory with that period length. The pentagons are also connected to "symmetric" and "asymmetric" to explain the difference visually. And it is "number of bounces" is connected to "combinatorial period length". In the video this diagram, or spatial story reveals itself one by one, starting with title and then with individual nodes and connections.
Liouville's Theorem revealed step by step in a structure of 10 interconnected nodes
black and white photo of myself with long hair, part of it covering my left eye and with gentle friendly smile