Far from being due to chance, the shapes of the universe are underpinned by geometric frames. We see their presence from atoms to clusters of galaxies, including plants, animals, the energy circuits of the Earth, and even the human body.

As we focus our attention on the plants and animals around us, we find that nature takes on a wide variety of shapes, which are usually rounded and rather irregular. Nothing seems to be geometric there. Yet, geometric structures underlie most of the natural forms.

**The unavoidable presence of polyhedra and spirals in nature**

It is this aspect that we recognize in crystals. When we observe calcite or quartz concretions, we find that they are delimited by regular plane facets. These facets intersect at sharp angles whose values are not arbitrary but determined by symmetries. The plane facets determine cubes, prisms, or other more complex volumes with plane faces called polyhedron. Such regular crystalline geometries are also found in many other minerals and in ice crystals.

Thanks to the discovery of X-rays in 1895 and to the discovery of the phenomenon of their diffraction by crystalline substances in 1912, scientists were able to determine that crystals were made up of a periodic and regular stack of elementary patterns composed of a few atoms. This pattern called mesh has the shape of a polyhedron, cubic, prismatic or parallelepiped. We have been able to count 14 different models of polyhedra compatible with regular stacking.

Although it seems less apparent, polyhedral forms are also found in animals and plants. These include the shells of turtles or crustaceans, the cuticles or eyes of certain insects. As for the flowers, many are those which bloom in a beautiful regular star. The stars are themselves polygons whose angles are re-entrant.

## In plants and animals, close observation reveals to us that there are many geometric shapes other than polyhedra.

Curved shapes are a second type of geometry abundant in nature. The spirals are the most remarkable representatives. We have examples of it in snail shells or ram’s horns.

In plants, their elements (leaves, flowers, thorns, scales) are staggered along the stem in an elaborate construction during growth. This construction is presented in two different modes. In one, the elements are arranged in groups of two facing each other. In the other, they are born one after the other and are placed in spirals all around the stem.

In the case of the spiral, the elements are placed with respect to each other at an angle whose value is 137.5 °. However, this value is the one which shares the circumference 360 ° according to the golden number 1.618.

For example, in a sunflower flower, the arrangement of the florets reveals many spirals. We can count 21 in one direction and 34 in the other, or 34 and 55, or 55/89 or 89/144. In a pineapple, you can find 8 rows of florets in one direction and 13 in the other. In the pinecone, 8 rows of scales in one direction and 13 in the other or 2 and 3, or 3/5, 5/8 or 8/13. In the daisy 21/34 and in the celery, 1/2. But all these numbers are exactly part of the sequence studied by the Italian mathematician Leonardo Fibonacci (around 1175 – around 1250): 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89. The golden ratio is reflected in the ratio of two of these consecutive numbers.

**Geometry in nature: fractals that follows patterns and are hazardous at the same time**

Tree-like structures, such as that of a tree, made of large branches which branch into smaller ones, which themselves branch into twigs, etc. can be described as fractal structures. It is a third type of geometry which does not resemble polyhedra or spirals.

Fractals are 2 or 3 dimensional synthetic images comprising sets of points, lines, or surfaces obtained by repeating a graphic or mathematical construction process. Each repetition step is associated with a reduction (or increase) in scale which gives the images obtained a fragmented, branched or porous appearance, which is identical whatever the scale of observation.

Underlying fractal structures are natural formations of irregular appearance. They are found in river systems, in the jagged form of mountains and in cloud formations. We discover it in the fragmented, cut, porous, filamentous material. The proof that a fractal structure is underlying these formations is that it is possible to simulate them graphically realistically by synthetic fractal images. This is for example the case with ferns, mountains, clouds and many other materials and landscapes.

We have shown that geometries, polygons or polyhedra, spirals and fractals, constitute an underlying framework of the forms of nature. In order to give a more realistic image, it is necessary to specify that this frame is not rigid. Plastic and alive, the geometry can undergo adaptations in relation to its ideal perfectly symmetrical shape. We can imagine this character by considering a tent which takes on wobbly aspects if it is installed on uneven ground. This does not take away from its manufacturing design with perfectly symmetrical and adjusted frames. Even if installed askew, its original perfect geometry is still evident in its apparent form.

Why do bees build hexagonal honeycombs? Well, they don't, they build irregular polyhedra which are hexagonal at one end. So, does this polyhedron minimise surface area & wax expenditure? No, it was proven in 1964 that it's suboptimal: https://t.co/26Pp5P1kwo @DoctorKarl pic.twitter.com/XUD7eFlnhS

— Gordon McCabe (@DrGordonMcCabe) November 4, 2020

**Exploring the Universe with numbers: basic mathematical considerations to analyze the sideral space**

Let us leave Earth and now elevate into sidereal space. The generally accepted representation is that of a universe expanding since the initial phase of the Big Bang. This could lead us to imagine that the stars are randomly dotted evenly throughout the cosmos, like pebbles in a field. However, it is not the case. Are we going to find geometric patterns there?

The billions of billions of billions of stars that inhabit the cosmos tend to agglomerate in clusters of tens or hundreds of billions of stars: galaxies. Between the galaxies, there are relatively empty spaces of stars. The first galaxy discovered was the one in which we are located, the Milky Way. Indeed, until the beginning of the 20th century, the Milky Way seemed to be the limit of our entire universe. It was only after the development of telescopes and spectroscopes that it was possible to estimate the distances which separate the Earth from visible stars. We could roughly estimate the diameter of the Milky Way.

In astronomy, distances are expressed in light years. A light year is the distance traveled by light in a year in empty space, at a speed of 300,000 km/s, or approximately 10,000 billion kilometers (300,000x60x60x24x365.25). It is therefore a unit of distance, not that of a duration. Astronomers also use parsec and its multiples. The parsec is a distance related to an angle of one arc second and is equal to 3.26 light years.

## The Milky Way is shaped like a thin disc swelling at its center.

Its radius is estimated at 45,000 light years. The solar system is outward, 26,000 light years from the center, or about 2/3 of the radius.

In 1923, Edwin Hubble (American astronomer, 1889 – 1953) discovered that the Andromeda Nebula is located outside the Milky Way, because he estimated its distance to 900,000 light years. However, it was recognized as being a galaxy itself. This is the beginning of the understanding that there are many galaxies outside of the Milky Way. To date, the number of cataloged galaxies is disproportionate.

Some galaxies have not a specific shape, but most take on geometric shapes. A small number are elliptical, others lenticular. Most galaxies are remarkably spiraling. Subsequently, astronomers realized that galaxies were not evenly distributed in the cosmos, nor were the stars within them. They are grouped into clusters.

Thus, next to the Milky Way, we find the Andromeda galaxy, the Magellanic clouds and about fifteen dwarf galaxies. This set is called the local group. Its size is 13 million light years (13 Mal), approximately 130 times the size of the Milky Way. Galaxies are assembled into groups, and groups are assembled into clusters that encompass both groups of galaxies and isolated galaxies. With an average size of 60 Mal, clusters are approximately 5 times larger than groups.

## But that is not all.

If we go higher in the scale of distances, we see that the clusters are not evenly distributed. Here they are grouped in super-clusters. A super-cluster includes about 5 or 6 clusters, and has an average size of about 260 Mal, which is 4 times larger than a cluster. For example, the local cluster containing the Milky Way is included in the Virgo supercluster, also called the local supercluster. It was discovered by the Franco-American astronomer Gérard de Vaucouleurs in 1960.

At this stage of our exploration, we have recognized 3 hierarchical levels of galaxy agglomeration (group, clusters and superclusters). Do you believe that this cosmic architecture stops at this scale? It is not, and the most amazing thing is yet to come.

This is because superclusters tend to cluster together in large, long, thin filaments, or pancake-like sheets. These sheets and filaments delimit large empty spaces, as if they were placed on the surface of empty soap bubbles. Cosmic void bubbles occupy a prominent place. They are estimated to be 650 Mal in size, but some filaments can be much longer. As for the vacuum, it is not empty, but populated by isolated galaxies and a tenuous gas.

With these bubbles, the cosmos takes on the appearance of a cellular structure. Cells (bubbles) are like cavities in a sponge, with galleries that connect the cavities. Therefore, we speak of the sponge structure of the universe. The cellular geometry of the universe was discovered in the 1990s (Broadhurst, Tully, Einasto) and surprised astronomers, because no model from the standard Big Bang theory could explain it.

**Interstellar geometry: what polyhedral, spirals and fractals can tell us about the Universe**

If we dare to better define the shape of cells in our near universe, we discover that the bubbles are not round. They have the specific form of polyhedra. This is the observation made in 1997 by a team of Spanish astronomers including E. Battaner and E. Florido. From maps incorporating the most recent data provided by satellites, these astronomers determined that super-clusters and intergalactic filaments occupied the vertices and edges of 4 octahedra that touched at the tip.

This structure cannot be explained within the framework of the usual models based only on the forces of gravity. On the other hand, if we add the magnetic fields to them, the calculations show that it is possible. But then, as these fields fill all the space, it is not only our near universe which would have this structure, but the whole universe. The universe would appear to be a stack of octahedral cells like an egg carton.

At the same time, from the same catalogs of galaxies, other astronomers analyzed the density of galaxies, clusters and super-clusters and they discovered that this density was distributed in a fractal structure. (eg Coleman and Pietronero 1992, Lindner et al. 1996).

## A structure is detected to be fractal by measuring its density in volumes varying over large scales.

For a homogeneous substance such as water, the mass contained in a cubic container increases as the size of the container to the power of 3. For example, if you fill a cube with sides 10 cm, you have one liter. If you fill a cube with a side of 20 cm, you have 8 liters (2x2x2 or 2 to the power of 3). It is different when the structure is fractal. Its density grows less quickly than the power of 3, according to a proportion whose power is between 1 and 3, and which is called its fractal dimension (see article Fractal images). The fractal dimension of the universe was evaluated at 2 by Coleman.

Battaner (1998) considers that this fractal character is completely compatible with the structure of cells in octahedra. It suffices to fragment each octahedron into smaller octahedra. For example, in an octahedron, there can be lodged 7 octahedra 3 times smaller which touch only at the tip. Mastering this method to analyze our Universe could lead us to a better comprehension of it and, therefore, help us to reach new frontiers.

Happy #DarkMatterDay 👻#Didyouknow that only ~ 5% of the #Universe is made of ordinary matter, like the one stars, planets & people are made of? 😱

We are looking forward to investigate the rest with our future #Euclid mission #darkmatter #darkenergy

👉 https://t.co/VtIywillwI pic.twitter.com/OwtPjjiys6— ESA Science (@esascience) October 31, 2019