An idea I have had about math that I find intellectually engaging is that math does not just help people weigh and measure things. Most people think of math as enabling them to find the area or volume of a geometric shape or determine how many miles to the gallon an automobile gets, but math actually describes relationships between things as well. For example, in a triangle, two of the three angles added together determine how big the third angle can be. The three angles are related to one another, and they depend on each other.
By the same token, math can be applied to concepts and objects that are not usually associated with numbers, such as beauty. A local weaver says that she determines the pattern in her handwoven scarves based on Fibonacci numbers, and the scarves have won awards for their beauty. Likewise, the dimensions of an individual's face determine its beauty. The mathematical proportions associated with the eyes, nose, and mouth are what make the face beautiful. A fraction of an inch wider or narrower, and a nose is not attractive. Lips not proportional to the face look overblown or too thin. Eyes that are mathematically too small, too close together, or too widely set are simply less beautiful than those that are mathematically perfect.
This idea that math does not just measure things but is an aspect of an object implies that math is everywhere and in everything. Every object is compared to a perfect model having perfect mathematical dimensions, and individual differences among objects relate to how they differ from that perfect model. The structure of objects is mathematical, and so is their level of beauty; relationships between features are mathematical as well. If you look at it this way, math is just a feature embedded in every object and every person that can be used to evaluate how beautiful or structurally sound they are.
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