Mathematics and more
Mathematics
Circumference:
2D Circle Circumference: 360° C = 2 r π
3D Circle Circumference: 540° C = 3 r π or C = 3 R / √2 π
Derivation:
r is the projected radius; R is the true spatial radius.
The 3D circle consists of six ¼ circles, so we have: C = 6 ¼ 2 r π » C = 3 r π
Arc: The 3D circle consists of six ¼ circles, and one ¼ circle has 90°, so we have 6 x 90° = 540°.
Area:
2D Circle Area: A = r² π
3D Circle Area: A = 3/2 r R π or 3/2 r² √2 π
Derivation:
Surface area of a cone: S = r R π
The circular region of the 3D circle consists of six ¼ cone surfaces, so we have: A = 6 ¼ r R π or 3/2 r R π
In the right-angled triangle: a² + b² = c², which corresponds to: r² + h² = R²
Since r = h, we have: r² + r² = R² » 2r² = R² » r² = R²/2 » r =√(R² / 2) » r =R / √2 and R = r √2
Therefore, the circular area of the 3D circle is: A = 3/2 r R π = 3/2 r (r√2) π = 3/2 r² √2 π
vortex
I am an architect, not a mathematician; otherwise, I would probably have found or at least explored many more purely mathematical applications. By chance, I came across Nikola Tesla's Vortex Mathematics, which is based on a circle with 9 points.
Here, one doubles a number (or halves it) repeatedly and then uses the digital root when the number exceeds 9. The number sequence 1, 2, 4, 8, 7 (digital root 16), 5 (digital root=32), (digital root 64=1) repeats infinitely.
When applying the same principle to the 3D Circle, one obtains the same result, with the difference that it becomes apparent that the numbers 1, 2, 4, 8, 7, and 5 are aligned at the same level, just as the numbers 3, 6, and 9 are aligned at a higher level.
What would Tesla have done with this information? What can modern mathematicians do with it?
Cardioid and Deltoid
I now know that by applying different numbers and multiplication factors using this method, interesting and aesthetic mathematical patterns can be generated, which are referred to in mathematics as 'Cardioid,' 'Deltoid,' and so on.
When using the same numbers and factors on the 3D circle, although an identical shape is obtained when viewed from above, in reality, a three-dimensional pattern is created by the spatial connection of the lines, which undoubtedly looks very aesthetic as well.
That's all for mathematics; I don't have anything more to say on the subject at the moment.
