The dodecahedron is a three-dimensional geometric figure that has 12 faces. This is its main characteristic, since the number of vertices and the number of edges can vary. Let's consider in the article the properties of this figure, its current use, as well as some interesting historical facts related to it.
General concepts about a figure
Dodecahedron - this word is taken from the language of the ancient Greeks, which literally means "figure with 12 faces." Its faces are polygons. Given the properties of space, as well as the definition of a dodecahedron, we can say that its polygons can have 11 sides or less. If the faces of the figure are formed by regular pentagons (a polygon having 5 sides and 5 vertices), then such a dodecahedron is called regular, it is one of the 5 Platonic objects.
The geometric properties of a regular dodecahedron
Having considered the question of what a dodecahedron is, we can go on to characterize the basic properties of a regular three-dimensional figure, that is, formed by identical pentagons.
Since the figure in question is three-dimensional, convex, and consists of polygons (pentagons), the Euler rule is valid for it, which establishes a unique relationship between the number of faces, edges, and vertices. It is written in the form: + = + 2, where - the number of faces, - vertices, P - edges. Knowing that the regular dodecahedron is a dodecahedron, the number of vertices of which is 20, then using the Euler rule, we get: P = T + B - 2 = 30 edges. The angles between adjacent faces of this Platonic figure are the same, they are equal to 116.57 o .
Mathematical formulas for the correct dodecahedron
Below are the basic formulas of the dodecahedron, which consists of regular pentagons. These formulas allow you to calculate its surface area, volume, and also determine the radii of the spheres that can be inscribed in the figure or described around it:
- The surface area of the dodecahedron, which is the product of 12 areas of pentagons with side "a", is expressed by the following formula: S = 3 * √ (25 + 10 * √5) * a 2 . For rough calculations, you can use the expression: S = 20.65 * a 2 .
- The volume of the regular dodecahedron, as well as its total face area, is uniquely determined from the knowledge of the side of the pentagon. This value is expressed by the following formula: V = 1/4 * (15 + 7 * √5) * a 3 , which is approximately equal to: V = 7.66 * a 3 .
- The radius of the inscribed circle, which touches the inner side of the faces of the figure in their center, is determined as follows: R 1 = 1/4 * a * √ ((50 + 22 * √5) / 5), or approximately R 1 = 1,11 * a .
- The circumscribed circle is drawn through 20 vertices of the regular dodecahedron. Its radius is determined by the formula: R 2 = √6 / 4 * a * √ (3 + √5), or approximately R 2 = 1.40 * a. The above figures indicate that the radius of the inner sphere inscribed in the dodecahedron is 79% of that for the described sphere.
Symmetry of the correct dodecahedron
As can be seen from the figure above, the dodecahedron is a fairly symmetrical figure. To describe these properties, crystallography introduces the concepts of symmetry elements, the main of which are rotary axes and reflection planes.
The idea of using these elements is simple: if you set the axis inside the crystal in question, and then rotate it around this axis by a certain angle, then the crystal will completely coincide with itself. The same applies to the plane, only the symmetry operation here is not the rotation of the figure, but its reflection.
The following symmetry elements are characteristic of the dodecahedron:
- 6 fifth-order axes (that is, the figure is rotated through an angle of 360/5 = 72 o ), which pass through the centers of pentagons located opposite each other;
- 15 axes of the second order (symmetric angle of rotation is 360/2 = 180 o ), which connect the midpoints of the opposite edges of the octahedron;
- 15 reflection planes passing through opposed edges of the figure;
- 10 axes of the third order (the symmetry operation is carried out when turning through an angle of 360/3 = 120 o ), which pass through the opposite vertices of the dodecahedron.
The modern use of the dodecahedron
Currently, geometric objects in the form of a dodecahedron are used in some areas of human activity:
- Dice for board games. Since the dodecahedron is a Platonic figure with high symmetry, objects of this form can be used in games where the continuation of events is probabilistic in nature. Most dice are made in a cubic form, since they are the easiest to make, but modern games are becoming more complex and diverse, which means they require dice with a lot of features. Dodecahedron-shaped dice are used in the Dungeons and Dragons role-playing board game. A feature of these bones is that the sum of the numbers located on opposite sides is always 13.
- Sources of sound. Modern speakers are often made in the form of a dodecahedron, because they spread sound in all directions and protect it from ambient noise.
History reference
As mentioned above, the dodecahedron is one of the five Platonic solids characterized by the fact that they are formed by the same regular polyhedra. The remaining four Platonic solids are the tetrahedron, the octahedron, the cube, and the icosahedron.
Mention of the dodecahedron dates back to the Babylonian civilization. However, the first detailed study of its geometric properties was done by ancient Greek philosophers. So, Pythagoras used a five-pointed star built on the tops of the pentagon (the edges of the dodecahedron) as the emblem of his school.
Plato described in detail the correct volumetric figures. The philosopher believed that they represent the main elements: the tetrahedron is fire; cube - earth; octahedron - air; icosahedron - water. Since the dodecahedron did not get any element, Plato suggested that he describes the development of the entire Universe.
Plato’s thoughts can be considered primitive and pseudoscientific by many, but this is curious: modern studies of the observable Universe show that cosmic radiation coming to the Earth has anisotropy (direction-dependent), and the symmetry of this anisotropy is in good agreement with the geometric properties of the dodecahedron.
Dodecahedron and Sacred Geometry
Sacred geometry is a collection of pseudoscientific (religious) knowledge that ascribes to various geometric figures and symbols a certain sacred meaning.
The importance of the dodecahedron polyhedron in sacred geometry lies in the perfection of its shape, which is endowed with the ability to bring surrounding bodies into harmony and evenly distribute energy between them. The dodecahedron is considered an ideal figure for the practice of meditation, since it plays the role of a conductor of consciousness in a different reality. He is credited with the ability to relieve stress in a person, restore memory, improve attention and concentration abilities.
Roman dodecahedron
In the middle of the XVIII century, as a result of some archaeological excavations in Europe, a strange object was found: it had the form of a dodecahedron made of bronze, its dimensions were several centimeters, and it was empty inside. However, the following is curious: a hole was made in each of its faces, and the diameter of all holes was different. Currently, more than 100 such objects have been found as a result of excavations in France, Italy, Germany and other European countries. All these items date back to the II-III century of our era and belong to the era of the Roman Empire.
How the Romans used these objects is not known, since no written source has been found that would contain an accurate explanation of their purpose. Only in some writings of Plutarch can one find the mention that these objects served to understand the characteristics of the 12 signs of the zodiac. The modern explanation of the mystery of Roman dodecahedrons has several versions:
- objects were used as candlesticks (the remains of wax were found inside them);
- they were used as dice;
- dodecahedrons could serve as a calendar, which indicated the time of planting crops;
- they could be used as a basis for fastening the Roman military standard.
There are other versions of the use of Roman dodecahedrons, however, none of them has accurate evidence. Only one thing is known: the ancient Romans highly valued these objects, since in excavations they are often found in hiding places along with gold and jewelry.