Timeline for Two circles and a pentagon
Current License: CC BY-SA 4.0
35 events
| when toggle format | what | by | license | comment | |
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| Jul 4 at 21:45 | history | edited | Trunk | CC BY-SA 4.0 |
MathJax improvements - used \text{...} to bring text onto same line as equation and save space
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| Jun 15 at 13:28 | history | edited | Trunk | CC BY-SA 4.0 |
Vain efforts at understanding blank line heights in concealed sections using >! etc
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| Jun 15 at 13:14 | history | edited | Trunk | CC BY-SA 4.0 |
\dfrac for \frac and other readability corrections, spacing, etc.
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| Apr 21 at 13:28 | history | edited | Trunk | CC BY-SA 4.0 |
More followability added
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| Apr 18 at 14:14 | history | edited | Trunk | CC BY-SA 4.0 |
Making algebra more followable
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| Apr 18 at 0:01 | history | edited | Trunk | CC BY-SA 4.0 |
New diagram 2
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| Apr 17 at 23:53 | history | edited | Trunk | CC BY-SA 4.0 |
New diagram
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| Apr 17 at 14:32 | history | edited | Trunk | CC BY-SA 4.0 |
Put digression in spoiler
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| Apr 17 at 12:58 | history | edited | Trunk | CC BY-SA 4.0 |
Better initial diagram
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| Apr 16 at 22:10 | comment | added | Trunk | @FirstName LastName Yet did it anyway as I couldn't follow neodne's neat solution. | |
| Apr 16 at 22:02 | history | edited | Trunk | CC BY-SA 4.0 |
Tidying up
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| Apr 14 at 0:32 | history | edited | Trunk | CC BY-SA 4.0 |
Wrote 5 as integer quotient directly comparable to 121/25.
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| Apr 10 at 23:49 | comment | added | Trunk | @FirstName LastName No - educational though this puzzle was to me, not another pentagon ! | |
| Apr 10 at 23:36 | history | edited | Trunk | CC BY-SA 4.0 |
Tidied fractions, etc
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| Apr 10 at 23:15 | history | edited | Trunk | CC BY-SA 4.0 |
Simple geometry-based proof of Cos 36 added
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| Apr 10 at 23:13 | comment | added | FirstName LastName | as in my answer to PSE#131103 : funny to spot 8/5 here (there 21/13) as rational approximation for golden ratio Φ where numerator and denominator are two consecutive fibonacci numbers | |
| Apr 10 at 23:02 | history | edited | Trunk | CC BY-SA 4.0 |
Simple geometry-based proof of Cos 36 added
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| Apr 10 at 13:48 | history | edited | Trunk | CC BY-SA 4.0 |
Added digression to explore pentagon geometry to find Cos 36 . . .
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| Apr 9 at 21:44 | comment | added | Trunk | Although the regular pentagon has enough geometric properties, e.g. overlapping parallelograms, to enable us to elicit the $\phi$ ratio between the diagonal and the edge length. Often wondered about US DoD obsession with this polygon . . . The USAF star, the DoD's very own HQ building, etc. | |
| Apr 8 at 13:05 | comment | added | Trunk | My point is that extracting $\cos{\frac{\pi}{5}}$ from the geometry of the 36-72-72 triangle is not at all obvious or as easy as say extracting $\cos{\frac{\pi}{4}}$ from the 45-90-45 triangle or extracting $\cos{\frac{\pi}{3}}$ from the 60-90-30 triangle. It is far easier to obtain $\cos{\frac{\pi}{5}}$ in the algebraic manner suggested by Oscar Lanzi, i.e. using the double- and triple-angle formulae. | |
| Apr 8 at 12:40 | comment | added | PM 2Ring | The golden ratio is everywhere in the pentagon. Both of the isosceles triangles it contains are "golden triangles", i.e., the ratio of the larger side to the smaller is the golden ratio, $\phi$. See cut-the-knot.org/pythagoras/cos36.shtml & cut-the-knot.org/do_you_know/GoldenRatioInRegularPentagon.shtml | |
| Apr 8 at 11:25 | comment | added | Trunk | @PM 2Ring a (36°, 72°, 72°) isosceles triangle (a point of a pentagram) with base 1 has the other two sides equal to the golden ratio ... This is the very part that isn't so obvious to me ! | |
| Apr 8 at 3:05 | comment | added | PM 2Ring | Trunk, a (36°, 72°, 72°) isoceles triangle (a point of a pentagram) with base 1 has the other two sides equal to the golden ratio. Now drop a perpendicular from the apex... | |
| Apr 7 at 11:31 | comment | added | Trunk | Handy, those proofs for exact values for multiples of $ \pi/10 $ and $ \pi / 18$ . I was casting about in vain for a geometrical means of getting $\sin{\pi/10}$ from an enclosed pentagon. I suppose it's there somewhere if you construct an elaborate diagram - but so far not found by me. | |
| Apr 7 at 9:54 | history | edited | Trunk | CC BY-SA 4.0 |
Adjusted size of graphic
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| Apr 7 at 1:18 | comment | added | Trunk | So many things a man should do . . . ! | |
| Apr 6 at 22:42 | comment | added | Oscar Lanzi | Much better. There are many derivations of the trigonometric functiobs of multiple if $\pi/10$ (or multiples of 18°, if you will) on Math SE and they are also given in Wikipedia. You should familiarize yourself with them. | |
| Apr 6 at 22:08 | history | edited | Trunk | CC BY-SA 4.0 |
Comment on 3-4-5 triangle occurrence.
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| Apr 6 at 21:59 | history | edited | Trunk | CC BY-SA 4.0 |
Amended "proof" provided.
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| Apr 6 at 21:59 | comment | added | Trunk | @Oscar Lanzi Alternate proof provided based on limiting case, i.e. $R_2 = R_1$. | |
| Apr 6 at 19:37 | comment | added | Oscar Lanzi | Reposted due to typos. I would try to incorporate exact trigonometric values derivable from the geometric properties of the regular pentagon. You get $\sin A=\sqrt{(5−\sqrt5)/8}$ and $\cos A=(\sqrt5+1)/4$. You then must prove with these radicals that $R_2>R_1$. | |
| Apr 6 at 17:33 | comment | added | Pranay | There are other answers with fewer calculations that don’t involve calculators. | |
| Apr 6 at 15:00 | history | edited | Trunk | CC BY-SA 4.0 |
Coda on extent of calculations
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| S Apr 6 at 14:16 | review | First answers | |||
| Apr 7 at 8:02 | |||||
| S Apr 6 at 14:16 | history | answered | Trunk | CC BY-SA 4.0 |