“although changed, I remain the same.” Epitaph of Jakob Bernoulli (1654-1705) in reference to the ubiquity of the logarithmic spiral in nature, the spira mirabilis. The size of the spiral increases but the shape is the same after each turn. The curve is said to be equiangular because from any point on the spiral, the angle is constant between a line drawn to the origin and a tangent to the point. Image: Klein Bottle, discovered as a four-dimensional mathematical (topological) object by Felix Klein in 1882, the Klein Bottle (“bottle” a misnomer) is like the Möbius Strip (also non-orientable) but with no surface boundary. The “self-intersection” said to be an immersion. Image: 2010-05-10_klein_bottle.jpg mediawingnuts.blogspot.com

eadum mutata resurgo