The Lemniscatory Geometry

Steiner introduces the lemniscate not as a stretched figure-eight but as one frame of the Curve of Cassini — the locus of a constant product of distances to two foci, just as the ellipse is the constant sum and the hyperbola the constant difference.

Sweep the focal distance below. The oval pinches into the lemniscate (K = c), then splits into two disjoint loops (K < c) — the case where, in his words, we are obliged “to go out of space… to preserve the continuity of the idea.” The dashed counter-space ghost stands in for that lost continuity.

The planetary loop and the human form are one lemniscatory principle. The figure closes upward into the head — the sphere pole, where Saturn, Jupiter and Mars have their loops in opposition — and opens downward into the limbs, the radial pole of Venus and Mercury, whose loops fall in conjunction. The Sun holds the rhythmic crossing between.

An eclipse is the crossing of two curves: the ecliptic — the Sun–Earth plane — and the inclined path of the Moon. They meet at two nodes, and when Sun, Earth and Moon align at a node, the crossing becomes an eclipse. (Inclination shown exaggerated for legibility.)

The whole system on this one geometry: Earth at the crossing node, a star field, the twelve-fold zodiac ring, and all seven spheres as nested Cassini curves. A single slider sweeps the entire system together through oval → lemniscate → split — the living metamorphosis behind the fixed loops you see in the Cosmos.