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Topic: BUSTED! Re: Myth-busting pentagons by compass & straight edge.
Replies: 9   Last Post: Aug 23, 2013 11:11 PM

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Jonathan Crabtree

Posts: 329
Registered: 12/19/10
BUSTED! Re: Myth-busting pentagons by compass & straight edge.
Posted: Aug 18, 2013 7:16 PM
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SHOW me don't TELL me! I want to try! You're too young to play with a compass Jonathan ...

"One of the most beautiful results in all of Euclid's elements is the construction of a REGULAR PENTAGON inscribed in a circle. (Book IV Prop. 11)"


OK I'm all grown up and at age 52, I actually believed all that hype about compass and straight edge construction of a regular pentagon (and everything else!)

Yes the proofs work. Yes the math works.

Yet published pentagons drawn by compass and straight edge (by which I mean intersection of arcs) ALL appear to be wonky, yes WONKY!

I know 'wonky' is not as precise as equilinear and equiangular, yet 'wonky' expresses my child-like disappointment felt upon discovering the truth about Santa and the Easter Bunny.

There seems to be growing circumstantial evidence that the construction of a REGULAR pentagon by compass and straight edge is simply too hard.

When a world-famous geometry professor draws a 'regular' pentagon with compass and straight edge (according to Euclid) and one side is 30 mm and another side is 37 mm - well, I'm ALMOST prepared to declare REGULAR (accurate) pentagon construction by compass and straight edge a MYTH.

So get out a ruler and CHECK FOR YOURSELF!


OK, so it's not enough to have one side 30 mm and another 37 mm, according to the legend of Euclid, this is, well, you read the PR spin...


I note the points around the pentagon are DUM NV.

So here we are 2300 years later with truck loads of algebra, trigonometry and dynamic geometry software and all I REALLY wanted to do was draw a regular pentagon with a compass and straight edge.

I bought quality bow-compasses and a steel straight edge.
I tried to follow Ptolemy and Euclid.
I failed.

So in order to SEE a regular pentagon be constructed, it appears we may need software.

The technology works* yet for me, the 'romance' of compass and straight edge construction has faded. Yet I fully expect others to maintain the faith...

Santa, Easter Bunny, regular pentagon construction with compass and straight edge...


No more picture story books for me! Much better to leave compass and straight edge construction to my imagination - or download some software or buy an app.

As the excellent book (containing wonky pentagon) referenced below reveals, Plato busted this myth 100 years before Euclid wrote the Elements.

"Although they make use of the visible forms and reason about them, they are not thinking of these, but of the ideals which they resemble, not the figures which they draw..."

So in conclusion, the myth is busted and my relationship with Euclid is now merely Platonic.

Jonathan Crabtree
Mathematics Researcher
Melbourne Australia
(Currently over at http://bit.ly/Vrrx9Q)

WONKY PENTAGON SOURCE: http://www.amazon.com/Geometry-Euclid-Beyond-Undergraduate-Mathematics/dp/0387986502/

* Watch Jennifer Silverman draw a pentagon with software https://www.dropbox.com/s/w9ckuc9fbujm16l/Pentagon%20Construction%201.mov

> Can a regular pentagon with five identical side
> lengths and five internal angles of 72° be inscribed
> on a circle with compass and straight edge only?
> Of course it can! Pentagons have appeared on coins
> for more than 2500 years.
> Countless geometers over thousands of years have
> drawn perfectly regular pentagons on a circle with
> compass and straight edge!
> Jonathan you are insane!!!
> ==============
> OK. It's just that when I check, the published images
> of hand drawn pentagons inscribed on a circle with
> compass and straight edge only are always, well,
> wonky!
> Have we enforced beauty on an irrational
> construction?
> I have tried to draw the perfect regular pentagon on
> a circle with compass and straight edge and FAILED!
> I have uploaded an early (slightly comical) video of
> me constructing a regular pentagon. It looks good,
> yet it's NOT regular!
> https://www.youtube.com/watch?v=sqE1s1oUBsU
> I can draw a regular pentagon around a clock face at
> 12 minute intervals. That's fun!
> I know the mathematics works for a regular pentagon,
> yet I have looked at Euclidean and other pentagons
> and followed instructions.
> We assume a regular pentagon can be drawn on a circle
> with straight edge and compass yet I can't and ALL
> the examples of published regular pentagons drawn by
> compass and straight edge on a circle have been
> wonky.
> And yes, I'm looking at Eu kid!
> So my request is, get out your geometry books and
> turn to the image of the pentagon hand-drawn with
> compass and straight edge on a circle.
> Measure the side lengths. Are they all consistent to
> 1 mm?
> Some I've seen are out by 1 cm yet said to be
> regular.
> Please give me proof they can be drawn. Don't TELL me
> SHOW me. Draw it! You can follow a method via this
> link...
> http://upload.wikimedia.org/wikipedia/commons/3/32/Reg
> ular_Pentagon_Inscribed_in_a_Circle_240px.gif
> If you draw a regular pentagon on a circle BY HAND
> (not dynamic geometry software) with straight edge
> and compass, EXACTLY, please email it to me and put
> me out of my misery.
> Otherwise I'll have to put in a request to
> Myth-Busters!
> Thank you,
> Jonathan Crabtree
> Mathematics Researcher
> Melbourne Australia
> (Currently over at http://bit.ly/Vrrx9Q)
> P.S. I can email examples of wonky pentagons on
> request. As some highly respected geometers are still
> alive, I do not wish to cause them embarrassment.
> P.P.S. Yes, I might be insane... But what if I'm not?
> My ruler has no vested interest.
> P.P.S. My 'comedic' YouTube video is unlisted and may
> be pulled shortly to spare me further grief!
> https://www.youtube.com/watch?v=sqE1s1oUBsU

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