Flat paper drops a dimension, and then we spend a lot of effort paying that debt back. A straight line, a perfect square, a sine wave drawn as up-and-down ridges: none of these exist in nature. They are projections. A sine wave is the shadow of circular motion. Trace that motion through time and you get a helix, and the wave is just its shadow on one wall. The complex exponential e^(iωt) is literally that helix, and the cosine is its projection onto a single axis. Engineers already trade waveforms for phasors for exactly this reason. So a lot of our "simplification" is really compensation for a flattening we performed in the first place.
We write the times table as a flat grid, but the product z = x · y is a real surface in three dimensions: a saddle, the hyperbolic paraboloid. Every cell of the times table is a height on that surface. The grid is the shadow; the saddle is the body. Once you can see the surface, you can read things off it that the grid hides: how it curves, the way it twists, where it is steep, and the diagonal ridge where the squares live.
Add the square and the surface changes into something that looks like a recliner, which is why I call it the SEAT. It is a real object with real geometry: angles, distances, inflection points, a half twist, and when copies line up along an axis they thread into a helix. Given only a definition of what you are looking at, you can derive attributes from the shape instead of storing them. Set the object inside a cube and each of the six faces and eight corners gives a different reading of the same thing.
In plain algebra, E = mc² is a member of the z = x · y² family: energy is mass times a factor squared, with z as E, x as m, and y as c. Kinetic energy ½mv² is in the family too, and so is a great deal of physics. The family is a lens. It lets you see at a glance which laws share a body.
If a relationship has a geometric body, you can compute its attributes from the shape rather than keeping them in a table: an angle, a distance, an inflection, a frequency along the helix. That is a representational idea, derive instead of store, and it is honest to say it is not a speed claim. In raw computation this is often slower than a lookup. Its value is consistency and derivation, one object read many ways, not throughput.
So the useful version of all this is geometry and representation: the multiplication surface, the SEAT, the helix behind the wave, the cost of flattening three dimensions into two, and the form as a unifying lens across formulas. The math is exact where it is math. The interpretation is a conjecture, offered openly and marked as one. That is the honest shape of an idea worth keeping.