Tuesday, January 02, 2007

Helix or Spiral?

I had the great privilage to visit Willi Kalender at the Institute for Medical Physics, University of Erlangen-Nürnberg. He welcomed me and invited me to hang my coat on a rather nice helix mounted on the wall with coat hangers on it. "By the way" he said "that is a SPIRAL coat hanger".

Prof Kalender is the father of a method of three dimensional computerized tomography he calls Multi-Slice Spiral CT, which greatly improved medical X-ray tomography. Relative to the patient the x-ray source describes a helical trajectory around the patient. Actually the source goes around in a circle relative to the hospital, and the patient moves through the machine in a direction normal to the plane of the circle. So why did he call it "Spiral"? I must add that Willi speaks excellent English, and checked his dictionary carefully. There are at least two instances in which "spiral" is used as a synonym for "helix". A "spiral stair case" and a "spiral bound" book. So he went for the more informal sounding word.

The Oxford Dictionary of English (revised edition).( Ed. Catherine Soanes and Angus Stevenson. Oxford University Press, 2005. Oxford Reference Online. Oxford University Press. Manchester University. 2 January 2007 t140.e74434)


winding in a continuous and gradually widening (or tightening) curve, either around a central point on a flat plane or about an axis so as to form a cone: a spiral pattern.

• winding in a continuous curve of constant diameter about a central axis, as though along a cylinder; helical. • (of a stairway) constantly turning in one direction as it rises, around a solid or open centre. • (Medicine) (of a fracture) curving round a long bone lengthwise. • short for spiral-bound: a spiral notebook.

Indeed the etymology stems from the Latin Spira meaning coil. However in modern mathematical English it is clear that a spiral is a plane curve and a helix a curve on a cylinder. Of course I am a mathematician, and also something of a pedant. So would avoid calling a mathematical curve by the wrong name, just as I would be careful not to say ball when I mean sphere, or linear when I mean affine, so I am likely to talk and write about CT with a helical source trajectory. However on the way back I mused that in mathematics it is usual for a great mathematician to invent something and call it something mundane while everyone else calls it after them. For example when Smale is talking about Morse-Smale dynamical systems he calles them "Simple" (with characteristic modesty). So I was thinking that we should talk of "Kalender Tomography", thus crediting the man who fathered the technique while remaining true to the mathematical terminology.

More links
  • Definition of spiral in Wiktionary. It agrees with me that spiral=helix is informal. No I didn't write it!
  • Wordinfo defines speira (Greek) spira (Latin) saying "The Greek speira refers to anything which is wound or wrapped around something" which sounds like it covers helices and spirals and "in the plural, speirai, it includes the twisted folds or coils of a serpent" which is even more general.
  • MathWorld helix, spiral.
  • PlanetMath circular helix, Arcimedean spiral