A twenty four hour clock has metric?

A spiral: ds^2 = (r dΦ)^2 + dr^2 where r=4Φ

Write everything in terms of Φ ; then r=4Φ, dr=4dΦ

ds^2 = (4Φ)^2 dΦ^2 + (4dΦ)^2 = (4dΦ)^2 (Φ^2 + 1)

ds = 2dΦ √(1 + Φ^2)

Integrating by parts, s = Φ√(Φ^2 + 1) + arcsinh(Φ) (+C)

But there's no point in that until you clarify the following:

>marks time every half-hour

=> Very ambiguous. Does that mean "one continuous revolution (Φ sweeps 2π) in half an hour"?

or that it is discrete and only advances on half-hour increments? and does Φ move 2π or (2π/(24*2)) in a half-hour? You need to clarify.

>What is the total distance traveled during odd intervals?

"odd intervals" of what? the hour digit? the half-hour increment? full revolutions of 2π?

EDIT: Al, it's not that nobody "got this", it's that you were utterly ambiguous! All you needed to do was answer the questions, dude.

So from the Wikipedia link, apparently you meant "a continuous 24 hour clock (=> Φ sweeps 2π in 24hrs, π/24 in a half-hour)".

But you STILL haven't answered what you meant by "odd increments":

1am to 3am? (odd increments in hours)

1am to 2am? (odd increments in half-hours)

something else? (odd increments of whole days?)

Try answering that this time, grrr.