How many pentagons are in a Dodecahedron?
The Dodecahedron is formed of 12 faces of regular pentagons.
It is made up from 20 vertices, 30 edges and the 12 faces. It is termed regular because each face is a regular polygon, in this instance that polygon being the pentagon.
In 3 dimensions 12 regular pentagons will fit nicely together to create this regular dodecahedra. This doesn’t happen in 2 dimensions, there will always be gaps if you use a regular pentagon to cover a flat surface.
What is a Dodecahedron?
“the sphere of twelve pentagons”
The Dodecahedron is one of the 5 Platonic solids. Named ‘platonic’ because it was Plato that described them in his writings. Plato composed Timaeus in approx 360 BCE as dialogues between the philosophers of the time, in it are descriptions of the geometric creation of the world which includes the five ‘platonic ‘solids.
There remained a fifth construction which God used for embroidering the constellations on the whole heaven
Plato’s Timaeus
The early Greek philosophers devised mystical geometric concepts that related to divine proportion as they knew it. The five Platonic solids are one of those concepts and curiously have persisted throughout history to still have significance in the science of today. They are more formerly known these days as ‘regular polyhedra’. Dodecahedra are one of these.
Each of these Platonic solids are made up of a certain number of surfaces all of the same shapes. The number of these surfaces leads to the name of the shape. Each form represented an element or state of matter. The dodecahedron was the fifth element or cosmos that held them all together.
- Tetrahedron is made of 4 triangles. – FIRE
- Octahedron is made of 8 triangles. – AIR
- Icosahedron is made of 20 triangles. – WATER
- Hexahedron is made of 6 squares. – EARTH
- Dodecahedron is made of 12 pentagons – COSMOS
The study of these forms in ancient times was the pinnacle of number comprehension as they knew it, both in terms of geometric and esoteric knowledge. Euclid’s book of Elements outlines the construction of these forms.
The thing about 12 pentagons and the dodecahedron….
12 is an abundant number
Like 6 circles fit snuggly around 1 in the 2D plane all touching the neighbours. So in 3 dimensions, 12 will fit around 1 all touching the neighbours and the centre. There are 12 directions out from the centre, which was seen by the ancients as all forces in balance.
It was this understanding of the twelve forces that come from the one in the centre that brought about the mirroring of this concept within structures of society. This would lead to the duodecimal system (made of multiples of six) which came to govern time and measures which we still use today.
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Symmetries of the Dodecahedron
The make up of the Dodecahedron gives us the recipe to work out the number of symmetries it holds. Dodecahedron is made of 12 pentagons, each pentagon has 10 symmetries : 5 rotational and 5 axial. The dodecahedron has 12 of these faces. 12×10 is 120.
The dodecahedron has 120 symmetries.
The other Platonic solid symmetries can be calculated using the same recipe. Symmetry of the plane shape x number of sides.
Carbon bonding replicates tetrahedral symmetry, crystals use cubed or octahedral symmetries and there is evidence of marine life structures being dodecahedral.
When we look with the scientists eye on the subatomic level there is evidence of these structure symmetries holding up the matter that makes up our lives. But of course we are not all scientists. And yet we don’t need to be. Simply being aware of the symmetries, however simple, and taking time to observe how they manifest in the environment around us takes us on a journey of discovery for the beauty of nature and our place within it.
The Universe in a Dodecahedron
When we are examining nature for signs of mathematical precision, we expect it to serve up Euclidean geometry. Our eyes have been trained for generations to look for the maths we studied at school and it seems hard for us to believe that nature can actually follow what appear to be man made patterning. Whereas it is man that has understood nature’s patterning.
It is these symmetries we need to pay attention to. Nature replicates the symmetry patterns of the Platonic solids. Nature works to symmetric structures. To the non-mathematician they may not be so easy to see. but if we start with the symmetries we know from school and begin to see the first steps of how they are manifest in natural forms we are well on the way to understanding the forces that create all matter.,
The Greeks claimed that the universe is a dodecahedron and contemporary study gives evidence they weren’t entirely off course. The symmetries we find around us today mirror exactly those of the Platonic solids.
Scientists can see these symmetries played out on all scales and with a little encouragement we can see them too,
Roman Dodecahedron
Roman dodecahedra appear to be a thing of curiosity. Some sort of tool from the Gallic Roman era made from bronze. An structure with 12 regular pentagonal faces each with a varying size circular hole in, revealing a hollow centre. Each vertex has a small circular ball attached giving an overal strange appearance. Speculation as to what exactly they were used for, but appear to be popular now for knitting accessory.
Dodecahedrons in Nature
There is a type of dodecahedron that you may well have eaten. Not the regular kind, but the irregular dodecahedron made of 12 Rhombic shape faces.
Rhombic Dodecahedron.
The pomegranate is an ancient fruit long associated with the prolific nature of life. For the greeks the pomegranate became a symbol of life’s regenerative nature and featured in the legend of Persephone and her ability to restore fertility to the earth.
The pomengranate displays its rhombic dodecahedrons beacuse of the dense packing principle. like the beehive’s hexagonal structure is the most efficient use of space and abundant produce. so too the spheres packed tightly will leave gaps, but ifthe seeds swell in ripe juicethey fsquish together to fill the gaps causing irregular rhombic dodecahedron jewels.
Pentagons, Dodecahedrons and the Divine Proportion
We know from our study of the pentagram star that it holds within it the divine proportion. Known as the Golden Mean, the small to the larger is equal to the larger to the whole. It is in this understanding that we fully grasp the concept of the Pentad and its life giving qualities
The pentagram star and the pentagon are one in the same. When we see one, the other is implied. So the pentagon holds the divine proportion too and by extension so does the Dodecahedron. A fact that the Greeks valued highly.
Being that the dodecahedron is made of 12 pentagons, the divine proportion is inherently present in the Dodecahedron. Using simple drawing techniques we can create the pentagonal patterns that lead us to the net of the 3D dodecahedral shape.
Any activity that uses pentagonal symmetry is preparing the way for an understanding of the Dodecahedron.
Craft Activities with Dodecahedrons
Dodecahedrons are surprisingly easy to make. We can draw a net to cut out and shape two ‘baskets’ of pentagons that fit together to make the dodecahedron. A PDF of this is on the Resource Library page In the Understanding Symmetries category for you to download.
And for those nimble fingers this origami dodecahedron brings our understanding of the 12 pentagons in a dodecahedron together very nicely. Click the collage image or go to Youtube
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Sources include:
- Beauty of numbers in Nature by Ian Stewart,
- The Secret Code by Priya Hemenway,
- Beginners Guide to Constructing the Universe by Michael Schneider.
- Mathematics for the Millions by Lancelot Hogben.
- String, Straightedge and Shadow by Julia Diggins.
- Teaching Mathematics with Origami ATM.
Through The Smart Happy Project Lisa communicates to her followers a voice of natural connection in a fast paced world. Following in the footsteps of philosophers and geometers of all ages Lisa embarks on highlighting our journey as humans in a natural world governed by patterns we can see and understand.
