the aligned distinct ring · sol mark construction

Newfound Mathematical 'Einstein' Shape Creates a Never-Repeating Pattern

A new shape called an einstein has taken the math world by storm. The craggy, hat-shaped tile can cover an infinite plane with patterns that never repeat.

Symmetric lenses | Proceedings of the 38th annual ACM SIGPLAN-SIGACT symposium on Principles of programming languages

A lens is a bidirectional transformation between a pair of connected data structures, capable of translating an edit on one structure into an appropriate edit on the other. Many varieties of lenses have been studied, but none, to date, has offered a ...

Hasharium — identities, given form

Enter a DID. Observe the form that was always inside it.

Quotient lenses | Proceedings of the 13th ACM SIGPLAN international conference on Functional programming

Bidirectional programming languages are a practical approach to the view update problem. Programs in these languages, called lenses, define both a view and an update policy - i.e., every program can be read as a function mapping sources to views as well ...

Sine, Saw, Square, Triangle, Pulse: Basic Waveforms in Synthesis and Their Properties

Learn the properties of 5 basic waveforms in sound synthesis to use their full potential in your synthesizer performance or design.

Aleph number

In mathematics, particularly in set theory, the aleph numbers are a sequence of numbers used to represent the cardinality of infinite sets. They were introduced by the mathematician Georg Cantor and are named after the symbol he used to denote them, the Hebrew letter aleph (ℵ).

Using spaced repetition systems to see through a piece of mathematics

By Michael Nielsen, January 2019

Unit Circle Trigonometry - haitham

An interactive sketch for visualizing sine and cosine functions with the unit circle

Circle of Fifths: The Key to Unlocking Harmonic Understanding

The circle of fifths is arguably the most helpful way of visually organizing the 12 chromatic pitches for learning music theory. Enjoy this lesson excerpt.

Signed Distance Fields – Render Diagrams

The virtuous circle

The virtuous circle is the most powerful pattern in developer tooling. And it's the myth of the 10x developer that makes it hard to see.

Calculating with lenses | Proceedings of the 20th ACM SIGPLAN workshop on Partial evaluation and program manipulation

Functional programs are particularly well suited to formal manipulation by equational reasoning. In particular, it is straightforward to use calculational methods for program transformation. Well-known transformation techniques, like tupling or the ...

Trees, maps, and theorems

Trees, maps, and theorems

Categories of Optics

Bidirectional data accessors such as lenses, prisms and traversals are all instances of the same general 'optic' construction. We give a careful account of this construction and show that it extends to a functor from the category of symmetric monoidal categories to itself. We also show that this construction enjoys a universal property: it freely adds counit morphisms to a symmetric monoidal category. Missing in the folklore is a general definition of 'lawfulness' that applies directly to any optic category. We provide such a definition and show that it is equivalent to the folklore profunctor optic laws.