Posts

Exploring Finite Fields, Part 4: The Power of Forgetting

Or: how I stopped learned to worrying and appreciate the Monad.

Exploring Finite Fields, Part 3: Roll a d20

When we extend fields with matrices, what other structures do we encounter?

Exploring Finite Fields, Part 2: Matrix Boogaloo

How do we extend a field non-symbolically?

Exploring Finite Fields: Preliminaries

How to do arithmetic when there is a finite number of numbers

Numbering Numbers, Part 2: Ordering Obliquely

How do we construct an irrational number from rational ones?

Numbering Numbers: From 0 to ∞

How do we count an infinitude of numbers?

Type Algebra and You, Part 3: Combinatorial Types

Isomorphisms through algebra on inducive types.

Type Algebra and You, Part 2: A Fixer-upper

Broadening arithmetic to exponentials and adding in fixed points.

Type Algebra and You, Part 1: Basics

Counting and arithmetic on finite types.

A Game of Permutations, Appendix

Partial Cayley graphs of Coxeter diagrams

A Game of Permutations, Part 3

Extending swap diagrams (with apologies to H.S.M. Coxeter).

A Game of Permutations, Part 1

Some basic, interesting connections between graph theory and permutation groups.

A Game of Permutations, Part 2

Notes on an operation which makes some very large graphs.

Further Notes on Algebraic Stereography

How do you rotate in 2D and 3D without standard trigonometry?

Algebraic Rotations and Stereography

How do you rotate in 2D and 3D without standard trigonometry?

Generating Polynomials Extra: Legendary

Another series of related, orthogonal polynomials

Generating Polynomials, Part 2: Ghostly Chains

What do polygons without distance still know about planar geometry?

Generating Polynomials, Part 1: Regular Constructibility

What kinds of regular polygons are constructible with compass and straightedge?

On the Volume of the Platonic Solids

How big are the Platonic solids in relation to one another?

12 Pentagons, Part 3

Exploring even more symmetries based on the 12 pentagons condition.

12 Pentagons, Part 2

There are 12 pentagons. Why stop at dodecahedra?

12 Pentagons, Part 1

There are 12 pentagons. How many hexagons can there be?

Of Infinitesimals and Exponents

Is there such a thing as an raising a number to an infinitesimal power?

Counting in 2D, Part 2: Reorienting Polynomials

Geometric properties of certain 2D systems and attempts to classify them.

Counting in 2D, Part 1: Lines, Leaves, and Sand

An introduction to two-dimensional counting using polynomials of two variables.

Polynomial Counting 5: Pentamerous multiplication

Arithmetic in non-geometric integral systems and surprisingly regular errors therein.

Polynomial Counting 4: Two 2's

Using irrational fractional counting systems to generate chaotic behavior.

Polynomial Counting 3: The third degree

Interesting results on simple irrational counting systems involving only zero and one.

Polynomial Counting 2: Binary and beyond

Continuing to explore irrational counting systems and discovering problems with certain polynomials.

Polynomial Counting 1: A primer

An exploration into generalizations of positional number systems to irrational numbers.