CIRCANGLE
Not everything is a circle. In fact most curves aren’t. π is just one value of the ratio of a curve to its radii, and only when the two radii are equal.
The book
Did the Ancient Greeks think that everything was triangles, then discover circles? Probably not, but Archimedes and Apollonius gave us the geometry, π came with it, and we then spent a couple of millennia pushing everything through a round hole. Well, I’ve done some hard maths (see inside) and found that π isn’t special. In fact, when it comes to curves, what you actually need is… triangles. This is my book on the circle triangle… the circangle. It’s time to ditch π as an obsolete format.
Recreational mathematics for the curious, with the hard maths left in.
What is a circangle?
A circangle is a symmetrical quarter of an ellipse, treated as a right-angled triangle: its legs are the two semi-axes a and b, and its hypotenuse is the curved arc. The circle is the case a = b.
π describes a curve only when its two radii are equal. The book replaces it with the complete elliptic integrals K(k) and E(k), which change with the shape and keep its information. π is simply their value for the circle: π = 2K(0).
Things to play with
Both run in your browser. Nothing to install, no sign-up.
The ellipse calculator
Enter two semi-axes and get the exact circumference and area, with the formulas worked through. See how far the π shortcuts are out.
Open the calculator →The circangle slider
Drag the modulus k from circle to flat line and watch the circangle, its right angles and K(k) and E(k) move together.
Open the slider →Inside the book
- The Problem, and Its ResolutionAn exact arc-length integral for the circangle (Robilliard’s Formula), why Ramanujan’s approximations take the form they do, and two new approximations.
- The Circangle’s ToolkitNear-golden ellipses, the circangle’s own area, π as the minimum of a circumference ratio, an angle measured by arc length, and the elliptic curve attached to every circangle.
- Extension: The SpherangleThe construction in three dimensions, giving the surface area and volume of the triaxial ellipsoid.
- Replacing πClassical formulas from geometry, finance, physics, chemistry, biology, optics and sound rewritten in circangle form, closing with the Millennium Prize Problems.
- A Practical Use of Circangle: The StellaratorThe twist a fusion stellarator gets from a rotating elliptical cross-section, and the smallest one that could burn deuterium and tritium with existing technology.
- Further ExplorationExplorations followed for their own sake.
215 pages · 7 × 10 in · colour figures · Kindle, paperback and hardback
Buy the book
Kindle
Fixed layout, every equation and figure exactly as printed. Read on Kindle devices and the free Kindle app.
Buy on AmazonThe author
James Robilliard is an independent mathematical researcher. Find him on LinkedIn.