The familiar trigonometric functions can be geometrically derived from a circle.
But what if, instead of the circle, we used a regular polygon?
In this animation, we see what the “polygonal sine” looks like for the square and the hexagon. The polygon is such that the inscribed circle has radius 1.
We’ll keep using the angle from the x-axis as the function’s input, instead of the distance along the shape’s boundary. (These are only the same value in the case of a unit circle!) This is why the square does not trace a straight diagonal line, as you might expect, but a segment of the tangent function. In other words, the speed of the dot around the polygon is not constant anymore, but the angle the dot makes changes at a constant rate.
Since these polygons are not perfectly symmetrical like the circle, the function will depend on the orientation of the polygon.
For the next round of discussion I’d like to shift the subject to the physical environment, posing the question, Is architecture rational?
Much of the newer work we see as we walk the streets of the city whether it’s New York, Seattle, Dubai, or the newer sections of Copenhagen, is more dramatic than architecture once was: taller, swoopier, twistier, less symmetrical. Architectural language, informed by the capabilities of parametric software and computerized fabrication tools, has become more fluid and less rectilinear.
From the onlooker’s perspective, it looks a lot like style. But when you talk to an architect, you often wind up having a conversation about how utterly pragmatic the building in question is.
For instance, the Seattle Central Library by OMA, completed in 2004. The lead architect on the project, Joshua Prince-Ramus, once told me: “Style freaks us out, the very word style.” He went on to explain the strange shape of the building—it looks like a monstrous mechanical jaw—by showing a diagram made by the library’s administrators of all the functions they required in the new building. Prince-Ramus claimed the architects translated the librarians’ chart directly into architectural form. He called this method “hyperrational.”
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