Rigid structures formed from elementary dodecahedra
The phenomenon of formation of symmetric crystal-like structures in the shape of regular and semi-regular polyhedrons
Rules of formation of structures referred to in the article:
1) Structures are formed from elementary dodecahedrons of the same size.
2) Adjacent faces of neighboring dodecahedrons are perfectly attached to each other with a complete alignment of vertices.
Generally accepted approaches to the construction of structures from polyhedrons imply the fullest possible filling of free space. The formation of cracks and gaps considered as a flaw and minimizing them constitutes the main purpose of packaging.
Described in the article method of constructing of three-dimensional structures is fundamentally different from the generally accepted approach in that the presence of side slots between the constituent elements of the structure is a necessary and desired condition. Packing of elementary dodecahedrons as dense as possible is not the major objective for our approach.
The construction of the structure begins with the central dodecahedron, by adding to it the outer dodecahedrons. One external dodecahedron to each of the twelve faces of the central dodecahedron. The outer dodecahedrons are held in place by mechanical bond with the central dodecahedron. As such, the mechanical bonding by abstract glue, having the same strength as the material of dodecahedrons, can be provisionally accepted.
As the layers of dodecahedrons add on, mutually coinciding and repeating geometric structures are formed. The specific shape of the “shells” of dodecahedrons formed by different layers is uniquely determined by the number of the layer of dodecahedrons in the structure.
The resulting structures are analogues of regular and semi-regular polyhedrons (Platonic and Archimedean solids). In particular, they are: Truncated Icosahedron, Icosidodecahedron, and a composite large Dodecahedron.
These unique structures are given a name FROIM, resembling FRAIM – obsolete form of frame.
We will begin the examination of the FROIM structures from a simple to complex. First structure is consisting of thirteen dodecahedrons: one dodecahedron in the center, and twelve surrounding dodecahedrons (one on the each face). The resulting structure has one layer around the central dodecahedron. Please take attention to the presence of gaps between the outer dodecahedrons. In this case, the central dodecahedron is completely blocked from the outside world; there are no gaps between the central and outer dodecahedrons.
Let’s add one dodecahedron to the outward-facing faces of the dodecahedrons of the first layer. We have formed a second layer of dodecahedrons. At this stage, we will not fill all free faces of the second layer, but will limit ourselves to the twelve most remote from the center upper faces, as that will allow us to obtain a rigid structure with the minimum possible number of dodecahedrons.
So far, in our construction consisting of three layers, twenty-five dodecahedrons are used (two layers of twelve dodecahedrons in each and one dodecahedron in the center). As before, there are gaps only between the side faces of the dodecahedrons, the axial faces have a perfect backlash-free fit.
Let’s expand our structure by adding fourth layer.
As can be seen from the figure, the fourth layer is added to the outward-facing side faces of the dodecahedrons of the third layer. To each of the 12 dodecahedrons of the third layer, five dodecahedrons of the fourth layer (60 in total) have been attached. The top faces of the third layer remain vacant. In this sense, the operation
to fill the fourth layer is the opposite of the operation to fill the third layer, where we added dodecahedrons to the upper faces, leaving free the side faces of the second layer.
Now in our design we have four layers containing a total of eighty-five dodecahedrons.
The dodecahedrons of the fourth layer formed pentahedral cells around each dodecahedron of the third layer.
And every three neighboring pentahedral cells formed hexahedral cells, in which two dodecahedrons from each pentagon take part. In general, the resulting figure resembles a classic Truncated Icosahedron.
The image of the truncated icosahedron is given for comparison on the right side of our four-layer FROIM. Classic truncated icosahedron has 32 faces: 12 pentagonal and 20 hexagonal. Four-layer FROIM truncated icosahedron also has 32 faces-sides: 12 faces made up of five dodecahedrons and 20 sides of the hexagons (made up of six dodecahedrons).
The four-layer structure is still not rigid enough; the dodecahedrons formed a tight connection at the points of contact with each other. But this contact is made only along the edge line of the adjacent dodecahedrons.
A much more rigid structure is formed with the addition of the next layer (the fifth).
To begin with, we will add only 30 (thirty) dodecahedrons to the existing ones in our structure.
The total number of dodecahedrons in a five-layer FROIM becomes 115. Obviously, there are many unfilled places where additional dodecahedrons can be placed, but we are now interested in the minimal possible structure that is most convenient for analysis. Let’s show that the resulting five-layer FROIM structure of 115 parts is similar to a Icosidodecahedron. A regular icosidodecahedron consists of 12 pentagons and 20 triangles.
For comparison, two images are presented:
On the left is separately reproduced upper (fifth) layer of our 115-element FROIM with translucent pentagonal planes superimposed on it. The dimensions of these auxiliary planes approximately coincide with the dimensions of the pentagonal structures formed by the dodecahedrons of the fifth layer. This technique helps to clearly visualize the general shape of the fifth layer of the resulting FROIM.
The gaps between pentagons have a triangular shape, as in the normal icosidodecahedron presented on the right for comparison. The number of triangular structures is also equal to 20, as in the classical icosidodecahedron.
Now we can talk in more detail about the rigidity of the resulting structure. The image below shows in enlarged view of the conjugation of dodecahedrons of the fifth layer (yellow) with the underlying dodecahedrons of the fourth layer (burgundy, purple and gray).
As you can see, the fit between dodecahedrons is perfect, there are no gaps. This fact suggests that the FROIM of the fifth order has the maximum rigidity with respect to
external pressure. This stiffness is determined by the stiffness of the individual dodecahedrons that make up the FROIM structure.
We continue to add the layers on our FROIM.
The 115 elements FROIM has 20 triangles; on each of them we place three dodecahedrons, which form the sixth layer of 60 dodecahedrons. Thus the simplest FROIM of the sixth level will consist of 115 + 20 x 3= 175 dodecahedrons.
The six-layer FROIM again resembles an ordinary icosidodecahedron, as it is composed of 12 pentagonal structures and 20 triangular. But pentagonal structures are not very obviously expressed, and triangular structures have smaller relative sizes as compared to pentagonal structures. But nevertheless, there is a formal similarity with a conventional icosidodecahedron.
As before, when we talked about a four-layer FROIM the structure of a six-layer FROIM is still not maximally rigid, the dodecahedrons formed a tight connection at the points of contact with each other. But this contact is made only along the edges of the adjacent dodecahedrons.
A much more rigid structure is formed with the addition of the next layer (the seventh).
The outer shell of the seven-layer FROIM is a giant dodecahedron composed of 20 elemental dodecahedrons. The total number of elemental dodecahedrons in the minimal set of a seven-layer FROIM is 195. This (again, as in the case of a five-layer FROIM) is a completely rigid structure, since the dodecahedrons of the last seventh layer ideally fit the dodecahedrons of the underlying sixth layer.
Ordinary classic polyhedrons are volumetric structures that are bounded by planes (flat shapes, polygons). The principal difference of the structures discussed in this article is that they do not represent a single closed volume, but consist of a set of interconnected individual volumes of the elementary dodecahedrons that together form structures of the appearance of regular and semi-regular polyhedrons. A kind of a “poly-polyhedrons”.
Since polyhedrons are composed of dodecahedrons that are in close contact with each other, the result is a mechanically stable structure. Layers of structures consistently change their external shape, depending on the number of the layer.
So down to the third layer, the structure retains the appearance of a dodecahedron. The next fourth layer takes the form of truncated icosahedrons.
Fifth layer takes form of an icosidodecahedron.
The sixth layer keeps the appearance of icosidodecahedron, but with different proportions than icosidodecahedron of the fourth layer.
The seventh layer returns to the shape of the dodecahedron, but having a size (approximately) 6.7 times larger than the elementary dodecahedron.
The main phenomenal property of FROIM structures is their rigidity. This is explained by the fact that the FROIM structures are characterized by a perfect fit between their component parts, that is, there are no gaps in the direction from the periphery to the center of the structure. Assuming that each individual dodecahedron is a rigid, incompressible body, we inevitably conclude that the resulting FROIM structures have rigidity equal to the rigidity of their constituent parts. Rigidity here refers to the ability to resist external pressure. In other words, FROIM structures are equally incompressible, as are their constituent elements. The condition for opposing external pressure is that external pressure must be applied strictly normally with respect to the center of FROIM structure (centrally symmetrical).
Full name of structures “FROIM” is (Phenomenal Rigid Object (of) Initial Matter), in this case, replacing the English digraph “Ph” with a more rational “F”.
I hope readers by this point have already caught the analogy of FROIMs with atomic nucleases. This analogy is especially evident in the quantitative matching of the constituent elements.
195 dodecahedrons FROIM structure
Layers from 2 to 7 are visible
We continue to analyze the properties of FROM structures. It is known that other regular polyhedra — cube, octahedron and tetrahedron-can be sequentially inscribed in a conventional dodecahedron.
Such a property is inherent in the structures, which we analyzing here.
Thus, the first structure is an analogue of the cube “inscribed” in the seven-layer “large dodecahedron”, which was presented in the previous section.
In the presented animation to facilitate the analysis it is shown only the upper four layers and Central dodecahedron. And the prototype cube inscribed in the dodecahedron is presented below for comparison.
Next in line FROM – the analogue of the tetrahedron:
Octahedron, more like a ball, and below its prototype – an ordinary polyhedron:
A more subtle version of the octahedron, devoid of most of the fourth layer dodecahedrons:
Another variant of the octahedron-like FROM structure, which differs from the previous one in the absence of the fifth layer dodecahedra:
And finally, the tetrahedron-like structure of the dodecahedrons, this time also four-layer:
Dodecagraf — atomic nucleus
Dodecagraf is derived from the word “dodecahedron” and “graf” – mathematical collection of sets (as usual, “f” instead of “ph”). Dodecagraf, or just graf.
In this section, we will present all the layers that can be formed from dodecahedrons by gradually increasing their quantity, starting with a single central dodecahedron.
We will distinguish rigid structures from ordinary non-rigid ones.
A rigid structures, so-called FROIMs is a Phenomenally Rigid Objects of the Initial Matter (see the appropriate section of this site). These structures provide strength to the whole structure of the nucleus, since they can not change its shape during collisions and under the force of external pressure.
Let us suppose that external forces are always applied centrally symmetric with respect to atoms. This is a logical assumption, since the outer atoms can be either other atoms (the maximum difference in the size of the atoms is less than 3 times), or the ether surrounding the atoms (applying the same pressure on all sides, which ensures the stability of the substance).
External forces are always directed to the compression of FROIM structures, since they are applied perpendicular to the touching faces of the dodecahedrons.
Conventional non-rigid structures from dodecahedrons are arranged at gaps inside FROIMs. Dodecahedrons of non-rigid structures can be separated from FROIMs with application of an external pressure or strokes. Since the external forces in this case are aimed at separating the dodecahedrons from each other.
All images are taken from the same distance from the camera to the Central dodecahedron. This should be considered when comparing the sizes of the components.
So layer 1 is the Central proton:
Layer 2 (12 protons located on all 12 faces of the central proton):
Since the Central proton is completely hidden from the outside world by side protons, in all subsequent structures we will not take it into account, that is, the total number of protons will always be reduced by one.
The first part of Layer 3 (rigid FROIM structure consisting of 3 layers):
Layer 3, fully filled — added 20 dodecahedrons between twelve dodecahedrons of rigid structure:
The previous image is completed with the first part of Layer 4 (rigid FROIM structure supplemented with 60 dodecahedrons):
Layer 4, fully filled — added 20 dodecahedrons (blue) between sixty dodecahedrons of rigid structure:
The previous image is supplemented with the first part of Layer 5 (rigid FROIM structure consisting of 30 yellow dodecahedrons):
The previous image was supplemented with the second part of the Layer 5 (FROIM a rigid structure consisting of 12 multi-colored dodecahedron the pentagonal centers of the rosettes):
The previous image is supplemented by the third part of Layer 5 — a structure consisting of 60 multi-colored dodecahedrons, 12 pentagonal rosettes:
The previous image is supplemented with Layer 6 (rigid FROIM structure consisting of 12 red dodecahedrons). Total number of dodecahedra (nucleons) 238 :
The Law of Mechanics 