User blog:Edwin Shade/An Intuitive Visualization Scheme for Ordinals Beneath ω^4

A few years yonder in the past I recall watching a YouTube video which featured a man who after suffering a concussion became a musical savant and was able to improvise complex pieces which he had not rehearsed beforehand. In the video he claimed he visualized music in the form of black and white squares. As a person who is fascinated by the workings of the human brain and loves learning, I since then have sought to visualize various mathematical formulas in the form of pictures, and is how, independently, I stumbled across the classic solution to the extraction of quadratic roots visually, or by completing the square. Though I cannot claim ability to the stunts certain savants are capable of, I can say that by relating the tangible with the intangible such as they have done, it affords me a wider field of view in which to examine a concept, and expands my horizons beyond one or two senses. The following notation is simple, but with that simplicity may afford one a wider horizon in which to look at ordinals, and to see past their merely symbolic forms and into their true nature, or the structures which they represent, as I have.

\(\square\) is equivalent to one.

\(\blacksquare\) is equivalent to \(\omega\).

\(\huge{\square}\) is equivalent to \(\omega^2\).

\(\huge{\blacksquare}\) is equivalent to \(\omega^3\).

Rules for square, (ordinal), manipulation.


 * A square concatenated with another square represents addition. (i.e. \(\blacksquare\square=\omega+1\)).


 * A square of a lesser value that is before another square in a term can be omitted. (i.e.\(\blacksquare{\huge{\square}}={\huge{\square}}\)), other wise the term may stay the same.


 * To add two terms of squares, concatenate them. (i.e. \(\blacksquare{\huge{\square}}+\square\blacksquare=\blacksquare{\huge{\square}}\square\blacksquare={\huge{\square}}\square\blacksquare={\huge{\square}}\blacksquare\))


 * To multiply together two terms of squares, concatenate the former term a number of times equivalent to the second term. If the second term is a limit term, (limit ordinal), then find the supremum of concatenation of the first string for all square term values beneath the given limit term.

That's all there is to it ! Any questions in the comments I will answer. In addition, I do plan on extending this visualization scheme far beyond \(\omega^4\); it is just that I feel more comfortable releasing the project little by little and hearing feedback from you, as the sole purpose of this is not as a mathematical notation, but as a really easy way to do ordinal mathematics in your head, and to combine the visual with the mathematical.