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.... we've created a new way to learn about geometry. Our main 'tool' is a hands-on color coded modeling system. Take them apart, put them back together, learn along the way. This is our project blog. Check out our about us or see more by clicking the square avatars of our host sites below. Got an icebreaker? Reach out with tumblrs' ask function. Or shoot us an email at newtoolslab@gmail.com. Cheers!

geodesic dome by letitiaf0x on Flickr. 8
iwannabe-billionare:

Hipsterrrrrrrrrrr blog
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Finally, we think inside the tetra. The volume inside is divided into 4 asymmetrical tetras. Again it shares the same base measurements, edge = X, base of 90 degree green triangle is 1/2X. The altitude is a constant green relationship 1/2X divided by the square root of 2 (1.14121356237309). The long side is 1/2X times the square root of 3 (1.73205080756888).

These four tetras that define the paths of least resistance between the center of each of them and the outside corners will be considered hyper-symmetric with each other. This means if you know one you know all of them. They have a self- organizing relationship among themselves. 

We must no longer assume that simple symmetry is more fundamental than asymmetry. Between the two is a deeper connection that looks asymmetrical on the surface but has a constant symmetric numerical relationship with others in the group.

It is more useful to think simplicity is a consequence of a primordial complexity than the other way around.

We found the constant symmetrical numerical relationships by measuring the 4 different colored 90 degree triangles. They are hyper- integrated with each other.

Any 90 degree triangle can be thought of as ‘one complete thought’, an integrity. The minimum definition of integrity is something that is in a state of completeness. An integral is a part necessary for the completion of the whole.

The root word for integrity and integral is integer meaning important numbers that define the integrals of an integrity. Integrity, integral and integer all grow out of the Latin word tangere that means to touch. 

Integration grows out of this same Latin root meaning the arrangement of parts into an integral whole. Hyper-integration defines a state of completeness with the maximum quality and speed of information circulating thought an integrity.

Finally, we think inside the tetra. The volume inside is divided into 4 asymmetrical tetras. Again it shares the same base measurements, edge = X, base of 90 degree green triangle is 1/2X. The altitude is a constant green relationship 1/2X divided by the square root of 2 (1.14121356237309). The long side is 1/2X times the square root of 3 (1.73205080756888).

These four tetras that define the paths of least resistance between the center of each of them and the outside corners will be considered hyper-symmetric with each other. This means if you know one you know all of them. They have a self- organizing relationship among themselves.

We must no longer assume that simple symmetry is more fundamental than asymmetry. Between the two is a deeper connection that looks asymmetrical on the surface but has a constant symmetric numerical relationship with others in the group.

It is more useful to think simplicity is a consequence of a primordial complexity than the other way around.

We found the constant symmetrical numerical relationships by measuring the 4 different colored 90 degree triangles. They are hyper- integrated with each other.

Any 90 degree triangle can be thought of as ‘one complete thought’, an integrity. The minimum definition of integrity is something that is in a state of completeness. An integral is a part necessary for the completion of the whole.

The root word for integrity and integral is integer meaning important numbers that define the integrals of an integrity. Integrity, integral and integer all grow out of the Latin word tangere that means to touch.

Integration grows out of this same Latin root meaning the arrangement of parts into an integral whole. Hyper-integration defines a state of completeness with the maximum quality and speed of information circulating thought an integrity.

Now we think inside the octa. The volume inside is divided into 8 asymmetrical tetras. The base remains exactly the same X and the mid-point 1/2X. The base and altitude of the blue 90 degree triangle of this tetra is 1/2X. The third side has a constant blue relationship of the square root of 2 (1.41421356237309) times 1/2X.

Now we think inside the octa. The volume inside is divided into 8 asymmetrical tetras. The base remains exactly the same X and the mid-point 1/2X. The base and altitude of the blue 90 degree triangle of this tetra is 1/2X. The third side has a constant blue relationship of the square root of 2 (1.41421356237309) times 1/2X.

Now we think inside the icosa. The icosa learned from the cubocta to define its paths of least resistance from the center to its outside corners using a asymmetrical tetra. The volume inside the icosa is divided into 20 of these tetras. These two tetras share the same base but the rest of the icosa tetra is a different constant relationship that is orange. We will call this orange triangle a bubble triangle. It is 1/2 of the four-sided bubble diamond.

Using the same method as before, we know the base has an X edge and the mid-point is 1/2 X. The altitude of our new 90 degree triangle has a constant orange relationship with its base, 1/2X divided by the golden ration as a number less than one (0.6180339887499). The third side of our new triangle (its long side) is 1/2X multiplied by the square root of 3.6180339887499 (1.90211303259031). This is the constant orange relationship.

You can think of the cubocta and the icosa as the original couple of all other structures. It is also practical to think of the octa and tetra as an original couple as well. 

It is my belief that the stability of basic structural order comes from a large variety of options. There is no One Way to organize the beginning of all structure. Maybe you have another way as well.

We now have enough information to compare and contrast the relative motion of the Jitterbug from the cubocta to the icosa. The long distance between the cubocta tetras is another 90 degree triangle colored blue representing the square. The square face of the cubocta has an edge of X. If you divide the square by the long axis it will have a constant blue relationship X multiplied by the square root of 2 (1.41421356237309). 

Contrast that with the new distance between the icosa tetras which is X. The square ‘morphed’ into 2 60 degree triangles. 8 plus 12 gives us 20 outside triangles of the icosa.

Now we think inside the icosa. The icosa learned from the cubocta to define its paths of least resistance from the center to its outside corners using a asymmetrical tetra. The volume inside the icosa is divided into 20 of these tetras. These two tetras share the same base but the rest of the icosa tetra is a different constant relationship that is orange. We will call this orange triangle a bubble triangle. It is 1/2 of the four-sided bubble diamond.

Using the same method as before, we know the base has an X edge and the mid-point is 1/2 X. The altitude of our new 90 degree triangle has a constant orange relationship with its base, 1/2X divided by the golden ration as a number less than one (0.6180339887499). The third side of our new triangle (its long side) is 1/2X multiplied by the square root of 3.6180339887499 (1.90211303259031). This is the constant orange relationship.

You can think of the cubocta and the icosa as the original couple of all other structures. It is also practical to think of the octa and tetra as an original couple as well.

It is my belief that the stability of basic structural order comes from a large variety of options. There is no One Way to organize the beginning of all structure. Maybe you have another way as well.

We now have enough information to compare and contrast the relative motion of the Jitterbug from the cubocta to the icosa. The long distance between the cubocta tetras is another 90 degree triangle colored blue representing the square. The square face of the cubocta has an edge of X. If you divide the square by the long axis it will have a constant blue relationship X multiplied by the square root of 2 (1.41421356237309).

Contrast that with the new distance between the icosa tetras which is X. The square ‘morphed’ into 2 60 degree triangles. 8 plus 12 gives us 20 outside triangles of the icosa.

To be more effective in “thinking outside the box” let’s think inside the box and contemplate the nature of the box itself. Think of these four basic structures as the four primordial boxes. They can be thought of as the most-effective-minimum expression of inside and outside of any box possible.

In this photo, we think inside the cubocta (cuboctahedron). There are 8 equal-edged tetras that define 24 paths of least resistance between the center and the 12 outside corners. The rest of the volume is defined by 6 half-octas.

For now, let’s think about what is constant on the outside of this interior tetra. The outside edge = X. Between two outside corners is a both/neither mid-point that equals 1/2X. That gives us the base of a 90 degree triangle. The third side of this triangle will always be the square root of 3 (1.73205080756888) multiplied by 1/2X. This is the altitude of the 60 degree equilateral triangle. 

This constant three-way relationship is given a yellow color. From now on, anytime you see yellow it symbolizes all of these constants.

To be more effective in “thinking outside the box” let’s think inside the box and contemplate the nature of the box itself. Think of these four basic structures as the four primordial boxes. They can be thought of as the most-effective-minimum expression of inside and outside of any box possible.

In this photo, we think inside the cubocta (cuboctahedron). There are 8 equal-edged tetras that define 24 paths of least resistance between the center and the 12 outside corners. The rest of the volume is defined by 6 half-octas.

For now, let’s think about what is constant on the outside of this interior tetra. The outside edge = X. Between two outside corners is a both/neither mid-point that equals 1/2X. That gives us the base of a 90 degree triangle. The third side of this triangle will always be the square root of 3 (1.73205080756888) multiplied by 1/2X. This is the altitude of the 60 degree equilateral triangle. This constant three-way relationship is given a yellow color. From now on, anytime you see yellow it symbolizes all of these constants.
Four “polyhedra” defined in the Jitterbug: 14-sided cuboctahedron, 20-sided icosahedron, 8-sided octahedron and the 4-sided tetrahedron.

After 25 years of model building, what does Fuller’s Jitterbug tell me about the interconnectedness of basic geometry?

Bucky Fuller was right, thinking does have a geometry and thoughts indeed have a shape. These are Fuller’s two main memes.  A meme is defined as a cultural item that is transmitted by repetition in a manner analogous to the biological transmission of genes. Thoughts and things share a common geometry.

The problem we face in understanding these two memes is the traditional language of fundamental structure lacks any clarity or precision when you try to couple both thoughts and things into one structural order, ‘one-complete-thought’, a “solid” thought. 

IF thinking has a geometry and thoughts have a shape THEN we need to crowd source a new language of basic structure. On this blog, the two most important qualities of this new language are:

Defining, measuring, mapping and building physical models of the primordial interconnectedness of basic structures. Primordial means constituting a beginning.

Defining, measuring, mapping and building physical models of the genealogy of initial structural growth. 

We can fold all of the complexity of our structural reality back to the very beginning of the idea of structure. All structures in our built environment can trace their ancestry back to the four structures of the Jitterbug.

It is my hope that each post can start a conversion with other people interested in crowd sourcing this new language. I will define new words that may be used as a ‘place holder’ to start the ball rolling.

Today two new memes:

The hyper-symmetry of the tetrahedron
The hyper-integration of the 90 degree triangle

Instead of using hyper as a prefix implying excess or exaggeration we will change its meaning to imply the most fundamental expression of its root word.

 So hyper-symmetry is a both/neither fundamental symmetry that exists at the mid-point between simple symmetry and asymmetry.

 Hyper-integration is the most fundamental form of integration that achieves maximum inter communication between the parts that are essential in defining a whole.

Four “polyhedra” defined in the Jitterbug: 14-sided cuboctahedron, 20-sided icosahedron, 8-sided octahedron and the 4-sided tetrahedron.

After 25 years of model building, what does Fuller’s Jitterbug tell me about the interconnectedness of basic geometry?

Bucky Fuller was right, thinking does have a geometry and thoughts indeed have a shape. These are Fuller’s two main memes.

A meme is defined as a cultural item that is transmitted by repetition in a manner analogous to the biological transmission of genes. Thoughts and things share a common geometry.

The problem we face in understanding these two memes is the traditional language of fundamental structure lacks any clarity or precision when you try to couple both thoughts and things into one structural order, ‘one-complete-thought’, a “solid” thought.

IF thinking has a geometry and thoughts have a shape THEN we need to crowd source a new language of basic structure. On this blog, the two most important qualities of this new language are:

Defining, measuring, mapping and building physical models of the primordial interconnectedness of basic structures. Primordial means constituting a beginning.

Defining, measuring, mapping and building physical models of the genealogy of initial structural growth. We can fold all of the complexity of our structural reality back to the very beginning of the idea of structure. All structures in our built environment can trace their ancestry back to the four structures of the Jitterbug.

It is my hope that each post can start a conversion with other people interested in crowd sourcing this new language. I will define new words that may be used as a ‘place holder’ to start the ball rolling.

Today two new memes:

The hyper-symmetry of the tetrahedron
The hyper-integration of the 90 degree triangle

Instead of using hyper as a prefix implying excess or exaggeration we will change its meaning to imply the most fundamental expression of its root word. So hyper-symmetry is a both/neither fundamental symmetry that exists at the mid-point between simple symmetry and asymmetry. Hyper-integration is the most fundamental form of integration that achieves maximum inter communication between the parts that are essential in defining a whole.
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This is part of what is inside the Jitterbug.

…Tom’s models

…Tom’s models

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Small traditional wooden models used in crystallography and mineralogy classes. A friend tipped me off to the similarities between these and Tom’s models and when I googled ‘mineralogy models’ the picture on the left came up. So similar!

Small traditional wooden models used in crystallography and mineralogy classes. A friend tipped me off to the similarities between these and Tom’s models and when I googled ‘mineralogy models’ the picture on the left came up. So similar!

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