Since ancient
times humans have wanted to transform materials, thus the early refinement of
iron, the creation of brass, and so on. In their quest to transform known
materials of their era, the ancients conceived the idea of elemental materials,
from which a compound material was created. Thus, it was correctly thought,
that if such a compound could be physically or chemically divided into its elemental units that
the material could be reconstructed into a different compound. The ancients
thus strove to identify the elements, and settled on the four elements of soil,
water and air and fire. About 350 BC, the Greek philosopher and
mathematician, Plato, believed that the universe could be explained from the
viewpoint of mathematics and he proposed five elements corresponding to five
perfect solids. The perfect solids are the tetrahedron (consisting of four
sides, each an equilateral triangle), the cube (six sides, each a perfect
square), the octahedron (eight sides, each an equilateral triangle), the dodecahedron
(12 sides, each an equilateral pentagon) and the icosahedron (20 sides, each an
equilateral triangle).
The five perfect solids
In the modern era the theoretical ‘fifth element’
is no longer a topic of consideration. However, we know that in previous
centuries and previous millennia scientists in both the Eastern and Western
cultures believed that five elements explained the universe. As to why this
belief in the existence of five elements was predominant -- regardless of place
or culture -- in the ancient and pre-Einsteinian eras, it is interesting and
perhaps pertinent to note that we also have five fingers, thus perhaps leading
to a sort of intuitive belief in the five-element concept. If we consider that
perhaps Plato had a transcendental cognition of the elements of the universe,
this could be the basis for his theory of the ‘magic’ element Aether,
which he believed was dodecahedron-shaped or possibly
another perfect solid of a shape which can be placed inside a dodecahedron. Although Plato believed that Aether has characteristics similar
to other perfect solids, he had difficulty in describing Aether in relation to
the other perfect solids, because at that time the Greeks believed that the
water element was sphere-shaped. Thus the icosahedron -- as the solid with a
shape most similar to the sphere – was related to water, while fire was related
to the edged tetrahedron, soil to the stable cube and air to the unstable
octahedron. Plato may have had only a transcendent cognition that Aether might
relate to a perfect solid, but without visualized evidence he could not relate
it to the solid in opposition to the common belief of the era.
Following this line of
thought -- and taking note of the fact that the dodecahedron has the same
symmetry as the icosahedron -- the authors of this
paper conceived that, as a mathematician,
Plato conceptualized
the inside of the dodecahedron mathematically.
An icosahedron can be generated from the lines which form the centers of the
faces of the dodecahedron (see illustrations, right). A mathematician would say that the dodecahedron and the
icosahedron are ‘duals’, as are the octahedron and the cube. Inversely, a
dodecahedron can be formed inside an icosahedron with the same method.
Similarly, a tetrahedron can be formed inside a larger tetrahedron; this arrangement
is referred to as a ‘self dual’. This means there is a possibility that, after
recognizing that the supposed ‘fifth element’ corresponded to a solid
consisting of 20 faces, Plato conceptualized an icosahedron inside a
dodecahedron. In 1871, the Russian chemist Mendeleyev organized the chemical
elements known at time into eight families on the periodic table.
Mendeleyev's periodic table (1871)At present, we know that there are 18 families
and two f-block families; correspondingly, we know that the chemical elements
can be classified into 20 families, thus relating to the 20-sided icosahedron. IUPAC periodic table of chemical elements (2007)
If we
contemplate two facts, one can find a similarity between the fifth element
relating to the 20 faces of the icosahedron and the 20 families of the modern
periodic table. In the other words, there is a possibility that, from his transcendental
meditation about Aether, Plato may have
intuitively known the element has 20 faces. Subsequently, modern chemists found
20 families in the elements from the viewpoint of chemical properties. Considering the above, we know of the 20 faces
in the icosahedron for Aether from the transcendental meditation of Plato, and modern
science divides the currently-known elements into 20 families. Conceptualizing what follows requires a jump in one’s thinking.
We are parts of the universe and our nature is the same to that of the
universe. This means that if one opens one’s mind philosophically, departing
from the viewpoints of traditional education and preconceived ideas, one can
see the nature of universe. This is the basis of meditation in Buddhism. In the
authors’ knowledge, Kant thought about this matter and he mentioned ‘the
transcendental knowledge’. Many inspirations of great scientific discovery are
achieved in this manner; in fact, Albert Einstein once said that he developed
his Theory of Relativity by visualizing himself sitting on a photon as it
dashed through space. The authors of this paper posit that Plato (and along
with him, Mendeleyev) also had a brilliant clarity of mind, enabling him to
visualize the structure of chemical elements. This brings us to a fundamental
philosophical question: “Can one find a thing which does not exist in the
universe?”
If there is a relation between the facts, there is a
one-to-one mapping between the icosahedron and the
periodic table.
Interestingly, the icosahedron has 10 faces in the middle band and
what are referred to as the ‘d-block transition metals’ can
be classified into 10 families. Thus the middle band of the
icosahedron directly
corresponds to the region of the ten d-block transition metal families, which a chemist would refer to as d
The
features of the icosahedron 3-D periodic table are as follows: 1. We can touch it. This
stimulates one’s interest in the elements, relating to chemistry as well as to science
in general, from young children taking their first look at chemistry to
professional scientists teaching the subject. 2. In the basic model, the
f-block elements are located in their correct position There
are many symmetry operations 3. One can handily
rearrange the periodic table depending upon one’s particular interest or experimentation. This is
the authors’ proposal for the next generation of chemists, teachers and
students. The authors posit that future generations will be able to change the
chemical properties of the elements and, correspondingly, they will change the
location of the families on the faces of icosahedron. This change, however,
does not mean they will be able to change the elements themselves as the
Alchemists attempted. In conclusion, the authors of this paper
believe that if there are other intellectual living things in the universe,
they too will begin their journey of scientific discovery with the recognition
of five perfect solids Footnotes: - We
should note here that these 14 elements are typically placed outside of
the standard two-dimensional periodic table. If we insert them in the 2-D
periodic table, the table is too wide. Moreover, the elements Lanthanide
or Actinide have similar chemical properties in the groups. They are
referred to as inner transition metals. They should be located between s
^{2}family (#2) and d^{1}family, #3.
- A ‘symmetry operation’
refers to an operation in which an object is of the same shape after the
operation as it had prior to that operation. A sphere, for example, has
infinite symmetry operations. An icosahedron features five symmetry
operations (rotating 360°/5 around the vertices), two symmetry
operations (around the center of faces) and two rotations around the center
of the edges. It also has five planar reflection operations for the
vertices and a point reflection operation for the center. The symmetry
operations usually simplify the description of the shapes of objectives.
Departing from the material science, we can also apply the concept of
symmetry to the numerous sets of two opposing relationships, for example,
positive and negative, left and right, good and bad, and so on. The
authors believe that a fundamental basis of our cognition is via symmetry operations.
In many cases, spoken language -- which expresses our cognition -- has
just two symmetry sets and a point in the center, thereby bringing some restriction
to understanding nature or society. This may stem from what can be
described as 1-dimensional or 2-dimensional education from youth onwards.
The 3-dimensional model of the periodic table can enhance one’s spatial
perception as well as provide multi-directional
view points for understanding the things around us.
- It
was proved there can only be five perfect solids in the Euclidian space. It
is fairly difficult to understand things in non-Euclidian space and we
have no experience in doing so. The authors assumed that Euclidian space
is one of the best models for (our) universe at this present time. The
point of this paper is generalization of the well-proven things to the
whole universe under the belief of the homogeneity of the universe and we
are just a part of that. Einstein
also believed and proved that the physical phenomena can be described as
the same form for the all observers. It is interesting to note that Mathematics
is a strange science; it starts only from a few assumptions but it clearly
describes the facts around us. The authors believe that others in the
universe also built up mathematics as humans have, step-by-step over time.
These others may find points, lines, 3-D solids and five perfect solids as
we have done. They can also find that the elements can be classified to 18
families. And, in this far away place, a strange man may correlate the
elements and families and posit that the periodicity of the elements can
be represented on the faces of icosahedrons in perfect logic(!).
- To explain the word ‘periodicity’ as it is
used in this context,
we should note that the elements show the chemical properties
*periodically,*as the atomic number of the element increases. Mendeleyev discovered this point. Periodicity is also a basis of our cognition. In modern science, it was believed that the material has wave-particle duality. Since the typical character of the wave is periodicity with space and time, the materials in the universe will show periodicity from the appropriate view points. We also simplified our observation with periodicity, just like symmetry. We express observable things with periodicity. One of the best examples is the measurement of time of in days, our life, the rotation of the earth and so on. We make units based on periodicity, for example, in linear terms, a ruler. In the chemical periodic table, the elements show similar chemical properties periodically as the atomic number of each element increases.
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