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Expansion to First 12 Geometrical Systems

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  The Expansion Includes 2 From the Bubble Family and
4 From the Sphere Family

In the 2. Guide to Basic Tool Kit portal, we discussed the two basic job descriptions for any fundamental geometrical system (polyhedron). The Bubble family (five-fold symmetries using Golden Ratio-related measurements) have a job of enclosing with an outside protective layer like an egg shell or cell wall. The Sphere family, when stacked together, fill all of the space without gaps. Some do this by themselves, some in pairs and others in a 3-way combination.

All size relativity now shifts to the 6BB VE, the outside limit of our You/Me Ball's center. Using Buckminster Fuller's theory of the closest packing of spheres, we define this 6BB = 12 cm as the maximum reach of the first cycle of spherical growth which builds a 6BB VE with 3 layers of closest-packed spheres around our You/Me Ball. The other 11 system of the "1st 12" will be relative to this limit.

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  The Bubble Family Additions
  Triacontahedron
      Icosidodecahedron
  30-Sided Triacontahedron
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We begin by tuning the size of the triacontahedron to fit inside (contained) by the 6BB VE. Multiply the long axis of the blue square face of the VE by GR (0.618033988749895...) and you have the length of the long axis of the triaconta, 10.48838...cm. The short axis of this bubble diamond is found multiplying the long axis by GR as well, 6.48217...cm.

Interconnectedness of the Bubble Family
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The 30-sided triacontahedron is unique among the most fundamental structures in basic geometry. 30 sides of the same shape is the maximum limit found in geometry. The triaconta is the outside layer for the Bubble family. The icosahedron and dodecahedron can be tuned to fit exactly inside it. The long axis of the triaconta's bubble diamond is equal to the edge of the icosa. The short axis is equal to the edge of the dodeca.

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The interconnectedness, synergy, of the Bubble family is demonstrated here. This is what I mean when I use the term 'a self-organizing' modular building pattern. The color coordination of the basic shapes defines their structural/numerical constants. If you know one measurement you know them all. These containment patterns can be easily replicated with other systems.

The long axis of the bubble diamond is 10.48838...cm which is the same length as the long axis of the dodeca and the edge of the icosa; synergy, behavior of the whole unpredicted by the behavior of the parts.

Triacontahedron's Equator
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You begin to construct the triacontahedron's equator by dividing the edge of the 6BB hexagon in multiples of GR. The 12 cm is divided into 7.416407...cm/4.58359...cm segments. From the corner of the internal triangles of the hexagon measure the 1.75077...cm lengths. You now have the lines of communication from the center out. All of these measurements have a 'handedness' to them meaning they have a spin. Here the spin is counter-clockwise. In the opposite hemisphere it is clockwise.

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Next draw a circle with a radius of 9.16718...cm, this gives you the outside limits of the equator. From the edge of the internal triangles of the hexagon measure 2.47597...cm and make a mark. Connect the two sides and you have the cone-like parts of the equator.

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To finish the outer limits of the equator connect the corners of the cones. That length is 6.93895...cm. The center line running through the cone-like shape has a length of 9.083207...cm, which is the altitude of the icosahedron's triangle in this series. The rest of the systems of the Bubble family fit inside the triaconta's equator.

62 Directions of the Triacontahedron

12 Red Directions

outside corners five around

20 Blue Directions

outside corners three around

30 Green Directions

center of bubble diamond

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  Icosahedron/Dodecahedron Combination Inside Triacontahedron
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At this size the icosa and dodeca with a shared center interpenetrate each other forming modular units that define the excess volume. The 'icosa-cap' is 5 60˚triangles with a pentagonal base. Its edge is half of the icosa edge. The 'dodeca-cap' is 3 asymmetrical triangles connecting the midpoints of the pentagon with a triangular base. There are 12 icosa-caps and 20 dodeca-caps. When you remove them you are left with the other new geometrical system in the Bubble family, the 32-sided icosidodecahedron.

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Equator of Icosahedron Inside Triacontahedron

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We first slice the icosa's triangle to accommodate the equator. There is a 'handedness' to these pieces. Tape two triangles together, measure 8.48528...cm along their shared edge to get an exact cut.

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The icosa's equator is built using two constants for the lines of communication between the center and the outside limits. The 9.083207..cm line of communication is the altitude of the icosa's triangle and is connected with the       (1.75077...cm) division of the hexagon's edge. The 8.48528...cm line of communication is the cut-off of the icosa's edge to accommodate the equator and is connected to the             (7.416407.../4.58359...cm) division of the hexagon. These two measurements will always be constant relative to the edge of the icosa.

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Equator of Dodecahedron inside Icosahedron

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Equator for Dodecahedron and Icosidodecahedron
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The equators for the dodeca and the icosidodeca are the same. This interior hexagon is tuned to 8.28528...cm. It is aligned with the GR multiples division of the larger 6BB hexagon edge. This location-in-common corresponds with the midpoint of the long side of the triaconta's equator and is the corner of the icosa's 12-sided dodecagon.

  32-Sided Icosidodecahedron
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The icosidodecahedron is the concave (inner most) layer of the Bubble family. It is the 'substructure' underneath the combination of the icosa and dodeca, thence the name. The triacontahedron is the convex outside layer, the icosa/dodeca combo provide the middle structure and the icosidodeca is the inner layer. The four of these systems combined into one outside layer provides an incredibly strong structural integrity for an outside protective shell.

On the far right, an alternative combination of VE and icosidodeca where they share the same axis of spin length. Variety of options is the basis of Design-in Nature.

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Here is another example of a 'self-organizing' pattern. If you know one measurement about the triaconta's bubble diamond you know the measurements of the icosa, dodeca and icosidodeca that fit inside the triacontahedron. There is a simple, logical symmetry among the interconnected numbers.

Equator of Icosidodecahedron
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62 Directions of the
Icosidodecahedron

12 Red Directions

center of pentagons

20 Blue Directions

center of triangles

30 Green Directions

outside corners

  Think Like a Sphere
VE/Icosidodecahedron Spherical Interconnectedness
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We know the distance between the center of the icosidodeca and the center of the square face of the VE is 8.48528...cm. This becomes the radius of our red circles. Six red circles can be scored and folded to construct an icosidodeca.  We can use the 90˚Triangle Formula to find (B) the altitude of the icosidodeca-cone. With this information we draw in the top and bottom cones on the circle with the edge of the icosidodeca = 5.24419...cm.

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We now connect the corners of the cone to the midpoint locations-in-common on the circle. That distance will always be the edge multiplied by the square root of 3.618... Divide that in half and you can draw in the other lines of communication to construct the 10-sided decagon.

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Repeat the same procedure on the opposite side. Find your technique to align these drawings. Above is the plan for scoring (cutting half-way through the material to act as a hinge) on the two sides.

  Four Circles Construct Spherical VE
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The icosidodeca and the VE have a symmetrical interconnectedness. In the second portal there is a detailed discussion on how circles self-organize to construct the 60˚mass production grid. Repeat the same drawing on the other side so you can score in the proper pattern.

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The Sphere and Bubble families have a deep symmetric synergy when you focus on physical model building. Whole systems thinking means you are constantly searching for the lines of communication between the essential parts of the whole.

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