If you view a spiral staircase along its axis, does perspective make it describe an equiangular spiral?

David Davis

I just spotted this stuff at David Thompson’s place, and I just wondered. Here’s a suspiciously equiangular one:-

All equiangular or logarithmic spirals follow the general formula

r = ae^(θ*cot k)

where r = radius of spiral at any cumulative angle from zero (radians), a is a coefficient, e is what “e” is, θ = the total of radians turned from r=0, and k is the “characteristic angle”, being the angle between the radius at any point and the tanget at that same point.

Such spirals describe the shape of galaxies, the flare rate of nautilus and mollusc shells, the patterning of sunflower seeds, and the flow of water out of a rotating sprinkler.

Entertaining table-talk for the LA dinner tomorrow!

6 responses to “If you view a spiral staircase along its axis, does perspective make it describe an equiangular spiral?

  1. All which point to the mind of the Creator; the greatest Mathematical Mind ever.

  2. Does the beauty of math point to the mind of a Creator? Hmm.

    Is there any conceivable universe; any purely hypothetical universe, in which the laws of mathematics are different? Start with the basics- can a universe exist in which 1 plus 1 does not equal 2? Can a universe possibly exist in which addition is not commutative?

    I would argue that such universes are intrinsically impossible. If there was a Creator, it was constrained to build universes in which mathematics is as we find it in our universe.

  3. There is an astonishing degree of order in the Universe. That’s why I get so hot under the collar at GramscoStaliNazis (let alone the real StaliNazis who are quite serious and don’t pussyfoot about with gradualist Fabian scumbag-schemes) who want to delay our comprehending of it all.

    I have never failed to state that /imo/ the task of all scientists is to help Men’s minds to “understand the Mind of God” – as Stephen Hawking put it (wish I’d thought of that phrase to describe the objective!) – although I note that he has backtracked recently on the God thing.

    I fall out with the creationists over their theses that anything seemingly complex must have had to have been designed deliberately to do what it does. All that was required was for the axioms of Physics to be what they are (and if they are even slightly changed, the Universe would end up different from what it is) at the “beginning”, and if allowed to proceed, then creation would unfold as it does and is doing.

    I tend to lean towards the position of the late Arthur Peacock, an Oxford Biochemist who wrote extensively about the (non- , as he saw it) the non-conflict between science and belief in an all-knowing God who exists through all time as well as before it and after it, personifies “order” (as in (i) John I ) and Knows what the outcome will be.

  4. Error: I meant COTANGENT (line 6 below the picture.)

    r = ae^(theta times cot k)

  5. That staircase does look very like a fossilised snail shell, if that answers your question.

  6. This one’s different:

    (Grand Hyatt, Shanghai)