# Counterintuitive

### The Rev. Canon Dr. James M. McPherson

Peano drew an arc that filled a square,

and Alexander’s Hornèd Sphere is wild:

the horns don’t meet, and yet they bridge the gap.

Infinity’s the Speewah of the mind:

the Realm of Odd, a Real of different kind

intriguing those who search with what they find.

NOTES

(1) Peano’s space-filling curve, Alexander’s Horned Sphere, and “wild”.

Peano’s space-filling curve and Alexander’s Horned Sphere are mathematical constructions which iterate to infinity with counterintuitive results.  Peano described the first “space-filling curve” in 1890, as a mapping of the unit interval onto the unit square (ie it passes through every point). He was motivated by Georg Cantor's earlier counterintuitive result that the infinite number of points in a unit interval has the same cardinality as the infinite number of points in the unit square.

The Alexander Horned Sphere is named after J W Alexander, who devised its iterative process in 1924. Mathematically, it is a way of embedding a sphere in 3-dimensional (Euclidean) space, which remains isomorphic (same shape) to a sphere; except that it has the curious counterintuitive property, that a loop around one of the “horns” cannot merely be slipped off it, the way it could be if the process were stopped after any finite number of iterations. Mathematically, “the sphere’s complement is not simply connected”. The horns’ convergence point is the sphere’s (only) “wild point”.

“Wild” and its antonym “tame” are the technical mathematical descriptors for types of embeddings of one manifold into another (a circle onto the surface of a sphere, a knot with no loose ends in three-dimensional space, etc). Loosely, “tame” is absolutely non-pathological, and “wild” suggests pathology at one point (“wild point”, “singularity”), at the least. The Alexander Horned Sphere is wild at the point where the horns converge towards each other without meeting (which is why the loop cannot be slipped off). The terminology of “wild and tame” occasioned much merriment amongst my friends when I was researching wild knots and arcs in algebraic topology.

(2) The Speewah.

Sometime in the last quarter of the nineteenth century, if not earlier, Australian yarn-spinners invented the mythical Crooked Mick, illustrious hero of the equally mythical Speewah – where the grandfather clock in the homestead hall had stood so long in the same place that the pendulum’s shadow had worn a hole in the back, and where the winters are so cold the mirages freeze over.

I have two collections of Speewah stories: Alan Marshall’s short compilation They were tough men on the Speewah; and Bill Wannan’s Crooked Mick of the Speewah (1965, revised 1985). Crooked Mick was also the subject of a 2005 movie which (according to its website) received a number of awards/nominations.

The location of the Speewah station itself is vague; it’s “way out west of sunset” (Wannan, 72). Alan Marshall (as relayed by Wannan, 80) quoted a correspondent who averred it was “originally the place a bit ‘farther out’, ‘over the next range’, where cattle were a bit wilder, horses a bit rougher and men a bit smarter than they were anywhere else”.

No matter where you start, Infinity is always that much further out along the track.