Trigonotrassic ordinal notation

Trigonotrassic Ordinal Notation (TTON or 2TON) is an ordinal notation created by Edwin Shade which uses triangles to build up patterns that equal ordinals.

Etymology

The name is from the Greek trigonon meaning triangle and the Middle English ‘truss’ meaning bundle, as in truss bridges, punned on with the word “Jurassic,” a reference to the time period the Stegosaurus lived in. This is fitting since this notation was made especially for the Stegosaurus Factorial, and the notation itself resembles both a stegosaurus’ back and bridge trusses.

Definition

Fig. 1

2TON is a “geometrically lexicographic” ordinal notation in that ordinals can be easily compared by seeing which structures come after other structures in the ordering. All structures are on a {3,6} tiling of cells which may either be empty or occupied. We order structures by measuring a portion of this tiling from a single triangle with its base parallel to the x-axis, called the oritrigonon (origin + trigonon). Every horizontal row of triangles is of descending priority from left to right, and the higher a row is from the oritrigonon the higher priority it has. (Higher priority means it’s closer to the front of the ‘alphabet’.) In this way we can order all structures contained within the “sextant” bounded by two extended lines of the oritrigonon in the {3,6} tiling. (A sextant is like a quadrant but one-sixth of the plane instead of one-fourth.)[See fig.1]

Triangles which have their vertex pointing up are called upurntrigons (up + turned + trigon) while triangles which have their vertex pointing down are called durntrigons (down + turned + trigons). Using this terminology, 2TON is defined like so:

• If no cells are filled except for the oritrigonon, the structure is equal to 1. [See fig. 2]
• If a structure is equal to the ordinal A, then the same structure with an upurntrigon filled in the next available empty cell in the oritrigonon row is equal to A + 1. [See fig. 2]
• The limit of a series of units which are all the same is one unit, then another unit but with a filled durntrigon placed in the first available slot in that unit, with every part of that unit after the filled durnitrigon removed. We tack on another upurntrigon at the end because a durntrigon can never precede an empty upurntigon. [See fig.3,4,5]

For practicality, we do not draw out the {3,6} grid every time but only the outlines of the filled cells, ignoring the empty cells.

Fig.2

Fig.3

Fig.4

Fig.5

The form of 2TON prior to ε₀ is isomorphic to single-row BMS, where (n) is converted into a solid row of n + 1 upurntrigons and n durntrigons between the upurntrigons. From ε₀ on new rules need to be introduced:

Fig. 6

• For a series of lengthening expressions with units U₁, U₂,. . ., the limit is U₁ followed by U₂ with an upurntrigon placed in the highest priorty slot on top of that unit. Note an upurntrigon must always rest on a durntrigon unless it is in the bottom-most oritrigon row. [See fig.6]

The units must increment by a constant structure, otherwise they do not count as a valid series of units over which one can find the limit.

[To be continued. . . . . . .]