Timeline for answer to Where is the pentagon in the Fibonacci sequence? by Prem
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| Jul 10, 2024 at 4:06 | comment | added | Prem | (1) Correct. That alternative way is nice too , @OscarLanzi , like what Yakk commented. (2) While your Suggestion will generate linear combination of $s$ & $d$ , my way & what Yakk gave will generate linear combination of $1$ & $x$ , Sum towards $\infty$ , Difference towards $O$ | |
| Jul 9, 2024 at 23:45 | comment | added | Oscar Lanzi | A similar construction using regular octagons, and choosing the unique set of diagonals parallel to the sides, generates Pell number coefficients. | |
| Jul 9, 2024 at 23:43 | comment | added | Oscar Lanzi | Here's what you may want to do. Let the first pentagon have side $s$ and the second have side $d$ (from the diagonal of the first pentagon). Show that subsequent pentagons have sides $s+d$, then $s+2d$, $2s+3d$, etc, generating the Fibonacci numbers as coefficients of the initial side and diagonal. | |
| Jul 4, 2024 at 15:01 | comment | added | Prem | This is Correct , @Yakk , while the ratio $Side::Diagonal$ is always Constant. Extending the Pentagons either towards $O$ or towards $\infty$ will involve FIB Series : Sum (Difference) of a term with a term having $x$ factor. | |
| Jul 4, 2024 at 14:34 | comment | added | Yakk | Some of the side-lengths of the blue pentagon in the bottom image might be illuminating. Namely, "base" has side length x-1, as do a few others. And, (x-1)+1 is x | |
| Jul 3, 2024 at 6:23 | history | edited | Prem | CC BY-SA 4.0 |
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| Jul 3, 2024 at 5:20 | comment | added | Prem | I know what you mean ,@QiaochuYuan , though my thinking is something like this : [[1]] Say we make a triangle with two Consecutive Sides , then calculate the largest internal angle ($108$) & then calculate the Diagonal (Side of the triangle) to get $2\sin(54)$ which will involve $\sqrt{5}$ , then we might ask : Where is the FIB Series ? We have used trigonometry to hide that relation which is then not easy to see. [[2]] What I have shown here is that we actually get the FIB Series relation via 2 "Similar" triangles. That will more easily highlight where / why the FIB Series is involved here. | |
| Jul 3, 2024 at 4:53 | comment | added | Qiaochu Yuan | The OP is already aware of this. The question is about the relationship to the Fibonacci numbers, not the golden ratio. | |
| Jul 3, 2024 at 4:51 | history | edited | Prem | CC BY-SA 4.0 |
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| Jul 3, 2024 at 4:39 | history | answered | Prem | CC BY-SA 4.0 |