The root3 value.

Dry Constructions

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Below is how I  came up with the root3 value:

I made the hexagram to be a unit length (by choice).

The circle will have a radius of  ¡Ì3/3 units (or 0.577350269 units)

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(done late-Oct.2003)

ONE DIMENSIONAL MEASURMENT

QUANTITIES e.g Perimeters

Expansion factor

(expanded/original shape)

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OBSERVATIONS

Shape

Original

Expanded

Hexagram perimeter

yellow: 6*1=

    6 units

orange: 6*¡Ì3 =

    6¡Ì3 units

¡Ì3

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All Length expand by ¡Ì3

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Circle perimeter

red:(2¦Ð*radius)=

    2¦Ð/¡Ì3 units

pink-red: 2¦Ð*1 =

     2¦Ð units

¡Ì3

Star perimeter

smaller star: 12*¡Ì3/3=

   4¡Ì3 units

bigger star:

12*1=

   12 units

¡Ì3

TWO DIMENSIONAL MEASURMENT

QUANTITIES e.g Areas

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Shape

Original

Expanded

Area of

Hexagram

yellow:  

    (3¡Ì3)/2  sq.units

orange:            

   (9¡Ì3)/2  

3

All Areas expand

by 3  or (¡Ì3² )

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Area of

Circle

red: ¦Ð*r*r =

      ¦Ð/3 sq.units

pink-red:

   ¦Ð sq.units

3

Area of

Star

smaller : 12*¡Ì3/12 =

      ¡Ì3 sq.units

bigger star:

   = 3¡Ì3 sq.units

3

THREE-DIMENSIONAL MEASUREMENT QUANTITIES e.g Volumes

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Shape

Original

Expanded

Volume of

Cube

yellow: 1*1*1=

   1 cu.units 

orange: ¡Ì3*¡Ì3*¡Ì3=

   3¡Ì3 cu.units

3¡Ì3

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All volumes expand by 3¡Ì3 or (¡Ì3³)

Volume of

Sphere

red:   4/3*¦Ð*r^3=

4¦Ð/9¡Ì3 cu.

pink-red:

= 4¦Ð/3 

3¡Ì3

Volume of.. uhm..Star

three-dimensional stars..?i*blink?!* ...on hold...

but this is expected: 3¡Ì3

SPECULATION:  n-DIMENSIONAL MEASUREMENT QUANTITIES

extrapolated:

¡Ì3ⁿ

Anything in here expands by ¡Ì3ⁿ

... ...

It has seemed to me that this expansion from one state to another also goes along with some rotation of either

30, 90, or 150 degrees (angle between a deep blue-line

and any of the pale blue lines from above diagram) creating 

wave-like patterns/spirals if traced...something like the figures below... but at the moment I can't go beyond this.

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I assumed that the smallest cubit expands into a bigger one and

through a rotation of 30deg. So I traced a point on the smallest cubit and followed it up to

the end. The result looks like some kind of an expanding spiral.

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A spiral resulting from a hexagram

rotating through 30deg. when expanding from one level to another immediate one.

..

 THE SECOND CIRCLE

  ..

The three of the six faces of the cube are made transparent so as

to view the sphere sitting perfectly in it touching the mid-points of the inside walls of the faces.

The touching is easily seen as it's also the meeting point of the light blue (cyan) lines. 

The cyan lines are included only for visual aid.

 This is the idea upon which the following (scroll down) comes into play .

'PERFECT ORIENTATIONS'

Seeing the cube and sphere from different view points 'sowing in easy fertile grounds'.

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Is this how a 3-D expansion of a cube and sphere looks like?

The sphere on another immediate level of a 3-D expansion would perfectly touches the 8vertices of the cube. This sphere would then sit into the cube on that new level..and so on....onto bigger levels...

THE SECOND CIRCLE

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This is what I was discovering:

The first circle touches the 6 vertices of the star of david/hexagram.

The second circle, I came to know of later, is that as seen from the sphere (fig.above) touching the 8 edges of the cube.

So herein came my dispute: since this first circle touches the edges of the hexagram and also the second circle/sphere touches the vertices of the cube then they should be the same and equal. I ranted on with this idea for some weeks trying to make them be the same but to no avail. The Hand calculations, as my last resort, showed that it isn't possible.

I found the ratio of the second to first circle as 2(¡Ì2)/3 (fig. below). The ratio easy to get with hand calculations. My scientific calculators can't keep up with roots. It changes the roots right from the beginning of the computations. So I'd have arrived at 1.2990381056766579701455847561294 which w'd have been so hard to me to simply it to 2(¡Ì2)/3.

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BELOW: HAND CALCULATIONS WITH VISUAL-SUPPORT
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Calculating the ratio  2(¡Ì2)/3

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Scaled

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A little explanation below of the above figure:

There are three stationery viewers A, B and C who are to work hand in hand to bring out all the facets of the cube and sphere while it changes into a 2-D hexagram and circle.

A is seated in the front eye view position (able to see Fig1,3,5)

B is seated in the top eye view position (able to see Fig 2)

C similarly in the left side eye view position (able to see Fig 4)

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Only two rotational processes are used in sequential order.

1. A  45deg. rotation of the cube

    Viewer A sees fig.1 mysteriously changing shape - elongating to fig.2

    Viewer B sees the real process of the 45deg.rotation but with no change in shape.

    Viewer C see the same thing as viewer A. (the majority A&C would think B is totally wrong)

2. A  35.26deg. rotation of the cube.

    Viewer A sees again a mysterious change of shape from fig.3. to fig.5

    Viewer B sees a mysterious change of shape (not shown on diagram).

    Viewer C sees the real process of the 35.26deg.rotation but with no change in shape.

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Emphasis is on how viewer A sees the final thing which is fig.5.

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Figure 6 in the above diagram

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This is the way I found out that there's a second circle(red) different from the first (light red/pink). It's ratio to the first one,I found to be 2(¡Ì2)/3.

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