Fibonacci series directly relates the tetrahedron to the pentagon | GEOMETRY: Platonic Solids & the Symmetries of Space | Sacred Geometry Web | Forum
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7:36 am
October 17, 2019
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If you take a tetrahedron and cut it with a plane keeping the three 60 degree angles at the top vertex but changing the lengths to any three terms of Fibonacci series the new fourth face is one point of the five-pointed star…all you need to solve are sine and cosine laws.
11:04 am
April 3, 2018
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That is interesting, but I’m not sure I understand fully.
Do you mean that, for instance, if the length of three edges is in proportions 3 : 5 : 8
then the new face would be a point of a 5 pointed star?
That seems counter-intuitive to me.
I can imagine that if the lengths were 3 : 3 : 5
that the new face might approximate a star point,
but “any three terms of the Fibonacci series” seems like it would often generate a new face
that was not an isosceles triangle, let alone a golden ratio one.
Perhaps I’m misunderstanding what you mean?
12:20 pm
October 17, 2019
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I actually do mean 3:5:8 but the phi ratio gets closer the higher in the sequence one goes; the 3:5 triangle has the shorter side of the star( the side of a pentagon) while the 3:8 and 5:8 triangles -dont ask me HOW…wind up having equal lengths for the third sides of their triangles- the two diagonals of the pentagon, or the sides that form the “point” of the star
12:37 pm
October 17, 2019
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and I should have clarified- the terms of the series must be in order, no “jumping around”
10:22 am
July 2, 2016
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I’m going to actually try this with sticks and glue.
Make a tetrahedron.
Mark points 3 5 and 8 units down three edges.
Join them up with other sticks.
Examine the resulting triangle.
I agree with @Cindy that intuitively I would think you need two lengths the same
in order to get an isosceles triangle, so Im intrigued 🙂
I find it fascinating that once you have phi ratios in a form
then they tend to appear everywhere within it.
So if the lengths were phi ratio instead of Fibonacci then I guess the triangle
would be a perfect star point?
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