The spiral and the double spiral
The Whirpool artwork
In 1957 Escher makes a curious drawing with a tessellation that follows a double spiral. Previously it had made many tessellations following circles, but more rarely following spirals. Probably the lack of symmetry of the last ones left him unsatisfied, and this was compensated by the double spiral, symmetric on both poles.
Whirpool (1957)
The spiral and the hyperbolic geometry
Apparently it has nothing to do with hyperbolic geometry, but the relationship is subtle. Escher saw the drawing in a paper by Coxeter.
Although some authors show this simply as a spiral drawing, the mathematics behind it are deeper.
Double Spiral on the Poincaré Disk
The duality between infinity and origin
The double spiral corresponds to an inversión of a regular spiral with respecto to a circle between the spires. This double spiral has two poles, one corresponding to infinity, which is mapped by inversion to the centre of the circle, and another corresponding to the center of the original spiral, which is mapped to the second pole, in a point within the circle.
The apparent symmetry of the two poles really correspond to the infinity and the center of the spiral. So it makes a philosophical interpretation about the duality of origin and infinity, returning one over the other.