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Topic: Spiral of Theodorus - unwound with inverse of integers
Replies: 3   Last Post: Sep 26, 2017 12:07 PM

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Mike Terry

Posts: 750
Registered: 12/6/04
Re: Spiral of Theodorus - unwound with inverse of integers
Posted: Sep 26, 2017 12:07 PM
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On 26/09/2017 12:11, wrote:
>> Yes it will converge to sqrt( (Pi^2/6) + 1) = 1.626325326...
> Oh wow! We can draw a circle with that radius and entire spiral will fit in! Nifty!

Yes, I'd not thought of it like that, but you're right. Also, looking
at the length of the spiral we get the series

1 + 1/2 + 1/3 + 1/4 + ... + 1/n + ...

This is another well known series called the harmonic series, and can be
shown fairly easily to diverge (i.e. sums become arbitrarily large as we
go on).

So... this implies the spiral circles the origin infinitely often, but
still fitting in the bounding circle.

I wonder if the spiral already has a name. (If not, we can call it the
Joshi spiral! :))


>> (i.e. the sum of inverses of square numbers) whose limit was first
>> evaluated by the famous mathematician Euler in 1734. It's called the
>> Basel problem, and you can search/read all about it in Wikipedia. The
>> infinite series sums to (Pi^2/6),

> Thanks. Will look up.

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