Definitely still awake,

(and counting tetrahedrons as they jump the fence).

Bisecting the 60 degree angle between the 3 and 5 lines intersects the 4 baseline at it's midpoint. If we draw a line (in the sand ! 😉 ) between the midpoint and the only uncovered side of length 8 where it just touches the sand we can see that there must be 2 equal length sides between the 3-8 and 5-8 points on the horizontal sand surface.

Just for interest (if anyone is still awake at this point?) the midline from the 4 baseline and the 8 side length point has a length of root 51 (square of 2 subtracted from square of root 55) or about 7.141 and the two equal line lengths are root 55 (8 squared - 3 squared) or about 7.416 and the shorter side = 4 as explained before.

The only snag is:

it's NOT a star point! The angle between the equal sides is approx 65 degrees not the 72 degrees in a star (360/5) while the short - long side ratio is approx 1.854 not Phi. 🙁

But it is definitely makes an isoceles triangle in the horizontal plane.

d0b123

]]>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?

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?

]]>It has at least 4 features that are each remarkable in themselves:

- It is a Platonic Solid yet it relates equally to the 3D cubeoctahedron, the cube, and the octahedron.
- It is the only shape in any dimension that tiles space like the square and 3D cube do, but is not cubic.
- It is the only shape in any dimension that is not tetrahedral and yet is its own 'Dual' - meaning that when truncated it produces another smaller version of itself.
- Its points are the same distance from their connected neighbours as from the centre of the design, and therefore it is the 4D Vector Equilibrium

Not only that but it is the 4D analog of Metatron's Cube.

These factors combine to make it one of the most intriguing forms in existence.

]]><br>

<br>Great Circle graph generator 2013-12-31

<br>http://freeman2.com/gcircle2.htm

<br>

<br>Complex number calculator 2013-03-28

<br>http://freeman2.com/complex4.htm

<br>

<br>Math Logic Truth Table 2012-12-08

<br>http://freeman2.com/logictt2.htm

<br>

<br>Build code to draw spherical coordinate

<br>http://freeman2.com/sphere3d.htm

<br>

<br>Spherical triangle graph 2011-07-10

<br>http://freeman2.com/sphtrian.htm

<br>

<br>Use Rodrigues Construction to prove

<br>Euler's Rotation Theorem 2011-07-03

<br>http://freeman2.com/tute0055.htm

<br>

<br>Proof of Newton's inequality

<br>http://freeman2.com/tute0043.htm

<br>

<br>Spelling check

<br>http://freeman2.com/dict000a.htm no dictionary, 102kBytes

<br>http://freeman2.com/dict67en.zip English, 1.88MBytes

<br>http://freeman2.com/dict34fr.zip French, 880kBytes

<br>http://freeman2.com/dict25es.zip Spanish, 654kBytes

<br>http://freeman2.com/dict53de.zip German, 1.78MBytes

<br>

<br>Javascript index has more programs

<br>http://freeman2.com/jsindex2.htm

<br>

<br>Confucius remarked: "While his parents are

<br>living, a son should not go far abroad. If

<br>he does, he should let parents know where

<br>he goes."

<br>

<br>Student TzuGong ask: "Is there one word

<br>which may guide one in practice throughout

<br>his whole life?"

<br>Confucius answered: "The word 'charity' is

<br>perhaps the word. What you do not wish

<br>others do to you, do not do unto them."

<br>2014-01-02-10-36 Liu,Hsinhan remark

<br>Confucius 551BC to 479BC.

<br>

<br>2014-01-02-17-16 Liu,Hsinhan 劉鑫漢 劉鑫漢 ]]>

Also it astounds me as it did many others in the late 19th century that we can actually conceptualize and draw the shadows of the geometries of higher dimensions, and that now we can cause our computers to rotate and morph these shapes in realtime.

Checkout this post for instance https://sacredgeometryweb.com/h.....nimations/

Perhaps the reason we can conceptualize these dimensions is that we actually DO have some experience of them ? As though for instance your point were actually creating a line as he moves through time, and as though there were other univerese where he moved through time in a different way and these created a plane ?

Now Im confusing myself - lets get back to basics... we are not points, and we do SEEM to perceive a 3 dimensional universe ... isnt it interesting to look at why that is so and what higher dimensions would be like ? My own experience is that it seems to be very clarifying and strengthening for my mind to even begin to be able to imagine such meta-structures!

See some of my glimpses of them here https://www.shapeways.com/shop.....eometryweb

]]>

Say we were in zero dimensionality talking to a zero dimensional creature. We could try and explain to him our world but he could never understand given he has no knowledge of height, length, or width. But lets say we are talking to a very bright dot who understands that if there were two of him right next to each other it would create a line. WONDERFUL! But that is still simply a theory of his own and he would understand length but height and width would remain a mystery until he was able to move into 1st dimension and contemplate on putting two lines together. I believe we are only able to see one level ahead to prepare for it. We could never truly understand 5th dimension until we have been to 4th dimension and experienced it. We are trying to understand higher level rules with 3rd dimensional habits.

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