# Tridentine Math

The Tridentine Mass is the Roman Rite Mass that appears in typical editions of the Roman Missal published from 1570 to 1962. — Tridentine Mass, Wikipedia

A 30°-60°-90° right triangle, with sides 1 : √3 : 2, can be divided into three identical copies of itself: 30°-60°-90° Right Triangle — a rep-3 rep-tile…

And if it can be divided into three, it can be divided into nine: …that is also a rep-9 rep-tile

Five of the sub-copies serve as the seed for an interesting fractal: Fractal stage #1 Fractal stage #2 Fractal stage #3 Fractal #4 Fractal #5 Fractal #6 Fractal #6 Tridentine cross (animated) Tridentine cross (static)

This is a different kind of rep-tile: Noniamond trapezoid

But it yields the same fractal cross: Fractal #1 Fractal #2 Fractal #3 Fractal #4 Fractal #5 Fractal #6 Tridentine cross (animated) Tridentine cross (static)

Elsewhere other-available:

Holey Trimmetry — another fractal cross

If you take a sheet of standard-sized paper and fold it in half from top to bottom, the folded sheet has the same proportions as the original, namely √2 : 1. In other words, if x = √2 / 2, then 1 / x = √2:

√2 = 1.414213562373…, √2 / 2 = 0.707106781186…, 1 / 0.707106781186… = 1.414213562373…

So you could say that paper has radical sheet (the square or other root of a number is also called its radix and √ is known as the radical sign). When a rectangle has the proportions √2 : 1, it can be tiled with an infinite number of copies of itself, the first copy having ½ the area of the original, the second ¼, the third ⅛, and so on. The radical sheet below is tiled with ten diminishing copies of itself, the final two having the same area:  You can also tile a radical sheet with six copies of itself, two copies having ¼ the area of the original and four having ⅛:  This tiling is when you might say the radical turns crucial, because you can create a fractal cross from it by repeatedly dividing and discarding. Suppose you divide a radical sheet into six copies as above, then discard two of the ⅛-sized rectangles, like this: Stage 1

Then repeat with the smaller rectangles: Stage 2 Stage 3 Stage 4 Stage 5 Animated version Fractile cross

The cross is slanted, but it’s easy to rotate the original rectangle and produce an upright cross:  