### Perhaps a master key shape for PHI

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#### Derek_Smith

• Fractal Fanatic
• Posts: 26

#### Perhaps a master key shape for PHI

« on: June 07, 2018, 07:51:04 PM »
Hi Phi Fans,

Some time ago the Harriss Spiral created some interest, so i thought i may try another shape for members to look at using geometry, link to Harriss Spiril below:-

https://www.geogebra.org/m/sdTJfqb7

We know that the Pentagon, Pentagram, decagram, decagon, and dodecahedron are all related and fit together like a glove, but only by using ten points in a circle not five.

The 3D dodecahedron appears by where the lines cross in lower diagram:-

https://en.wikipedia.org/wiki/Decagon#/File-9-simplex_t0.svg

Scroll down above link to Petrie Polygons and tap on 9-Simplex to make larger.

However this will show two opposite pentagons etc. etc.

If we can double up, lets try the Golden Spiral and mirror image, the shape that would give is a Cardioid, be careful i know what you may be thinking, let me finish my observations first, i have to show how i fit all the other shapes into the Phi spiral and mirror image, it turned my brain into custard for years, but now i think i can do it, i'll continue soon slowly, i'm getting better after breaking my spine!  Link to Golden Spiral below:-

https://en.wikipedia.org/wiki/Golden_spiral

Use a face mirror under the spiral to see Cardioid.
« Last Edit: June 07, 2018, 08:01:55 PM by Derek_Smith »

#### Fraktalist

• Strange Attractor
• Posts: 1063

#### Re: Perhaps a master key shape for PHI

« Reply #1 on: June 07, 2018, 08:29:50 PM »
hey!
yep, you guessed right, I was thinking exactly that and it was one of the first thing i tested with phimatrix. attached is a screenshot - sadly it doesn't match..

#### Derek_Smith

• Fractal Fanatic
• Posts: 26

#### Re: Perhaps a master key shape for PHI

« Reply #2 on: June 07, 2018, 09:06:41 PM »
I'm an astronomer, so was aware of the Retrograde motion of Venus that traces a Cardioid and a pentacle, however obviously an illusion by Earth and Venus in orbit around the Sun but having different orbit speeds.

However this did not help my master key as there are other ways to draw a cardioid.

I'll leave a link to show how two circles rolling around each other create the image:-

http://mathworld.wolfram.com/Cardioid.html

I'm getting tired, so will continue tomorrow,

#### Derek_Smith

• Fractal Fanatic
• Posts: 26

#### Re: Perhaps a master key shape for PHI

« Reply #3 on: June 07, 2018, 09:19:53 PM »
Hi Fractalist,

I'm hunting a link that isn't right at the moment, i'll continue our interesting debate tomorrow, i had guessed what you say!

#### Derek_Smith

• Fractal Fanatic
• Posts: 26

#### Re: Perhaps a master key shape for PHI

« Reply #4 on: June 07, 2018, 09:58:21 PM »
The link i was searching for regarding Venus retrograde motion is below, hope it works now, i must rest my back, i will be back tomorrow:-

#### Derek_Smith

• Fractal Fanatic
• Posts: 26

#### Re: Perhaps a master key shape for PHI

« Reply #5 on: June 08, 2018, 12:53:20 PM »
Hi Fractalist,

There is more than one way to draw a Cardioid, the other way is by a Cardioid Envelope, which can be by any number of circle envelopes, however Mathworld by chance show the right number, please scroll down link to Cardioid Envelope, you will notice that swirling circles pass through the inner circle at ten equal points.   From this we can add, two opposite pentacles, two opposite pentagons, and a 3D dodecahedron at center.

http://mathworld.wolfram.com/Cardioid.html

Obviously the inner circle gets divided into ten, thus we can then draw the decagon matrix shown below on link, scroll down to "9 simplex":-

https://en.wikipedia.org/wiki/Decagon#/media/File-9-simplex_t0.svg

This includes most shapes of Phi within one shape, i wonder if members would agree if it has value as a new shape of the Golden Ratio?

#### Derek_Smith

• Fractal Fanatic
• Posts: 26

#### Re: Perhaps a master key shape for PHI

« Reply #6 on: June 13, 2018, 02:54:46 PM »
The Golden Decagon Matrix showing two opposite pentagons,pentacles decagon, decagram and 3D dodecahedron is below:-

https://andreblackscribe.deviantart.com/art/Golden-Decagon-matrix-279545770

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