Theorems About Hectagons

[SaxonViolence]

I saw the long running thread about "Non-Figurate Numbers" and finally got curious enough to look up what they were.

 

I should have done that sooner.

 

Sorry Turtle. :unsure: :(

 

Anyway it jogged my memory about something that I once read.

 

If I remember correctly it was Descartes who was disturbed by the Fact that while no one can picture a Hectagon (100 sided Regular Polygon) in their mind's eye—it simply appears as a Circle...

 

That nonetheless Geometers of his day had just Discovered and had Proven some interesting Theorems about Hectagons.

 

{Using Deduction of Course}

 

Does anyone happen to know what those "Interesting Theorems" were?

 

 

 

Saxon Violence

[Turtle]

I saw the long running thread about "Non-Figurate Numbers" and finally got curious enough to look up what they were.

I should have done that sooner.

 

Sorry Turtle. :unsure: :(

 

I hadn't noticed. No harm, no foul. :mellow:

 

 

Anyway it jogged my memory about something that I once read.

If I remember correctly it was Descartes who was disturbed by the Fact that while no one can picture a Hectagon (100 sided Regular Polygon) in their mind's eye—it simply appears as a Circle...

That nonetheless Geometers of his day had just Discovered and had Proven some interesting Theorems about Hectagons.

{Using Deduction of Course}

Does anyone happen to know what those "Interesting Theorems" were?

 

Saxon

 

So, it's hectogon and I find no reference to Descarte or any 'new' theorems. In fact after 45 min of searching I find even the top math sites repeat the same meme.Wolfram MathWorld:

A 100-sided polygon, virtually indistinguishable in appearance from a circle except at very high magnification.

 

There is no special uniqueness about 100-gons as far as seeing/discerning a Circle. Obviously n-gons of more than 100 sides would be equally indistinguishable from a circle as is a hectogon, but it is also true of some n-gons of fewer sides. The attached image is a centered gnomon of 89-gonal numbers. All of the rings are 89-gons.

All-in-all, nothing to get hung about. Scale is as scale does. :circle:

[SaxonViolence]

O yes, the Hectogons weren't Unique.

 

They just served to Illustrate to Descartes _{(or was it Mozart; Lenin or maybe Stan Lee...)} some of the limits of Knowledge/Personal Experience.

 

But apparently Pre-Victorian Society was all abuzz about recent advances in Geometric Theory, which caused Descartes to really dig into the old Metaphysical/Epistemological Foundations of Science, Math and Philosophy of Mind.

 

And just for curiosity sake, I wondered what the Theorem was...

 

I know!

 

A 100-Gon will have exactly One More Side than a 99-Gon—no more than one more and no less than one more.

 

HMMMmmnn......?

 

My Euclidean Proofs are a bit Rusty.

 

I might have to cheat and say "Definition" in the "Justification" Column.

 

 

Saxon Violence

Edited August 18, 2013 by SaxonViolence

[JMJones0424]

I think you are looking for chiliagon, a 1000-sided object. Descartes wrote about them in his Meditations.