The dimensions of the Texas apron, as specified in Grand Lodge Law, provides a solution for the ancient problems of "Squaring the Circle."
The dimensions of the Texas apron are based on the Phi Proportion (or Golden Section), which is approximately 1.61803.... The Texas apron is sixteen inches square, with a six-inch drop in the triangular bib. This divides the height of the apron into two parts, the bib (6 inches), and the part from the tip of the bib to the bottom of the apron (10 inches). Think of drawing the apron on a checkerboard, which is 8 squares by 8. If each square is 2 by 2, then the checkerboard is 16 by 16. The tip of the bib would be at a point 3 squares down from the top on the vertical centerline of the checkerboard.
If you take a compass, and put one point at the tip of the bib, and the other point at the mid-point of the bottom edge of the apron (or checkerboard), so that you have a radius of ten inches between the compass points, and then draw a circle, using the point at the tip of the bib as the center of the circle, you will have a circle with a circumference that very closely approximates the perimeter of the apron.
In fact, each of the two bottom edges of the bib will also approximate the radius so that the circle will just touch the top two corners of the apron (or checkerboard).
An equilateral triangle can be drawn between the two bottom points of the apron and the tip of the bib. The dimensions of this triangle are based on the Phi Proportion. The height of that triangle, when divided by half the base, will approximate 1.618.... (The Phi Proportion). That triangle is of the same dimensions as the Great Pyramid on the Gizeh Plateau in Egypt.
The Phi Proportion can be demonstrated by the Fibonacci series of numbers: 1, 1, 2, 3, 5, 8, 13, 21 ..., where each number in the series, added to the next number, will give you the third number. So, 3 plus 5 equals 8, and 5 plus 8 equals 13, and so on. When you take any number in the series and divide it by the previous number, the answer will approximate 1.618... (The Phi Proportion). So, 21 divided by 13 equals 1.615.... The higher you go up the number series, the ratio of one number to the previous becomes closer and closer to the exact Phi Proportion.
Here's an interesting factoid: Years ago, when archaeologists were stumped by the fact that there seemed to be an expanding ratio of distances along the "Avenue of the Dead" in the Teotihuacan Pyramid complex near Mexico City, they asked the Astronomer General from Scotland to take a look at their drawings. He did some quick measurements and realized that the distances were based on the Fibonacci Series, and therefore, the Phi Proportion. The archaeologists, not being mathematicians, did not understand what he was trying to tell them. So, he picked up a guitar and laid the neck of the guitar along the blueprint drawing of the Avenue of the Dead. He then fretted the guitar neck at each of the points on the drawing and played the subsequent chords, each one higher in pitch than the previous. Everyone immediately recognized the music "Thus Spake Zarathustra" by Richard Strauss. This was the same music used in Stanley Kubrick's movie "2001: A Space Odyssey" when all the planets came into alignment.