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Classification of solids, Platonic solids
While growing up, we develop an approach of the space around us which helps us navigating and moving around the different objects in the space. These objects, these solids can be classified by several different aspects.
There exist solids, which are bounded by only polygons. These are called polyhedra. These shapes have the following illustrative property: if we are to produce a polyhedron made out of rubber, then we would be able to fill it up with air such that it would become an empty sphere. Two well-known example for polyhedra are the cube and the cuboid.
If a polyhedron satisfies the following property: for the plane of any face, every point of the polyhedron lies in one half-space determined by the plane, then the polyhedron is called a convex polyhedron.
There exist solids bounded by plane figures and curved surfaces. For some of these, their surface can be laid in a plane (for example the cylinder), and for others whose curved surface cannot be laid in a plane (for example the sphere). (Figure 19)
can be laid in a plane
\latex{ a }
\latex{ a }
\latex{ r }
\latex{ r }
\latex{ r }
Figure 19
According to this classification, the surface of solid can be determined as follows.
For polyhedra, the surface is the sum of the areas of the bounding polygons. If the surface of a solid can be laid in a plane, then one has to measure the area of the laid surface. If the surface of a solid cannot be laid in a plane, then determining the area of its surface requires more advanced mathematical knowledge.
Another classification can be given by concentrating on the method of construction for solids. For our studies in high school, it is enough to define the following three large groups.
- Cylindric solids can be created by following a plane figure by a line pointing in a fixed direction and having one single common point with the plane of the figure. Then the lateral surface obtained this way is intersected by a pair of planes parallel to the plane of the starting plane figure. (Figure 20)
m:attitude
top face
a:generatix
base face
Figure 20
For the line that ran around the plane figure, the segment between the top and base faces is called the generatrix, and the distance between the top and base faces is called the altitude.
If the base of the solid is a polygon, then it is called a prism, if circle, then it is called a cylinder.
If the live which ran around is perpendicular to the base plane, then the solid is called a right cylindrical solid, otherwise it is called an oblique cylindrical solid.
The right prisms with base being a regular polygon are called regular prisms.
The right prisms with base being a regular polygon are called regular prisms.
- Conical solids are created by moving a line along the border of a plane figure such that it is incident to a fixed point (the apex or vertex) outside the plane of the plane figure. (Figure 21)
The segments connecting the apex with the points on the border of the base face are called generatrices, the distance between the vertex and the plane of the base face is called the altitude. If the base of the solid is a polygon, then it is called a pyramid, if it is a circle, then it is called a cone.
m:attitude
generatrix
Figure 21
If every generatrices of a cone are equal in length, then it is called a right cone, otherwise an oblique cone. In the case of a right cone, the angle between the two generatrices determined by an axial intersection of the cone is called the aperture. (Figure 22)
aperture
right cone
oblique cone
\latex{ m }
\latex{ m }
Figure 22
Pyramids whose base is a regular polygon and whose edges are equal in length, are called regular pyramids.
Tetrahedrons are pyramids with base being a triangle. A tetrahedron is called regular only if all of its faces (and not only its base face) are regular.
Tetrahedrons are pyramids with base being a triangle. A tetrahedron is called regular only if all of its faces (and not only its base face) are regular.
- A sphere is the set of the points equally far from a fixed point in space. We can obtain a sphere by rotating a circle around one of its diameters. (Figure 23)
\latex{ r }
\latex{ 0 }
Figure 23
◆ ◆ ◆
Polyhedra can differ a lot in terms of its faces, edges and vertices. Suppose that every face of a polyhedron is made out of rubber. If any face can be made into a disc without tearing the polyhedron, then it is called simple.
By observing a few of this type of polyhedra one can discover a specific connection between the number of faces, edges and vertices. A few examples are shown in the table below:
Tetrahedron
Cube
Hexagonal
pyramid
pyramid
Number
of vertices \latex{(v)}
of vertices \latex{(v)}
Number
of faces \latex{(f)}
of faces \latex{(f)}
Number
of edges \latex{(e)}
of edges \latex{(e)}
\latex{ 4 }
\latex{ 8 }
\latex{ 7 }
\latex{ 4 }
\latex{ 6 }
\latex{ 7 }
\latex{ 6 }
\latex{ 12 }
\latex{ 12 }
Starting with a simple polyhedron, one can construct a planar representation – a graph – which can be obtained by “pressing it into” the plane keeping the vertices and edges. These graphs are called the Schlegel diagrams of polyhedra. This process is shown for the cube in Figure 24.
These planar graphs can be used to prove the following equality, which is called Euler's polyhedron formula:
These planar graphs can be used to prove the following equality, which is called Euler's polyhedron formula:
\latex{v+f=e+2}.
Polyhedra whose faces are congruent regular polygons and whose edge angles and dihedral angles are equal are called regular (or Platonic) solids.
Euler's polyhedron formula can be used to discover the different types of regular solids. Let every face being a regular \latex{ n }-gon. Since each edge is incident to two faces,
Euler's polyhedron formula can be used to discover the different types of regular solids. Let every face being a regular \latex{ n }-gon. Since each edge is incident to two faces,
\latex{f\times n=2\times e}.
If there are \latex{ k } edges incident to every vertex, then (since both endpoints of every edge are vertices):
\latex{k\times v=2\times e}.
Using these equations, it follows that
\latex{f=\frac{2e}{n}}; \latex{v=\frac{2e}{k}}.
Figure 24
By Euler's theorem, After dividing both sides by \latex{ e } we have: This means that Using equivalent transformations this yields
\latex{\frac{2e}{n}+\frac{2e}k=e+2}.
\latex{\frac2 n+\frac 2 k=1+\frac 2 e}.
\latex{\frac 2 n+\frac 2 k\gt 1}.
\latex{2k+2n\gt kn},
\latex{(n-2)\times(k-2)\lt4}.
\latex{(n-2)\times(k-2)\lt4}.
Obviously, \latex{ n } and \latex{ k } cannot be smaller than \latex{ 3 }, thus the factors \latex{n – 2} and \latex{k – 2} in the above inequality are positive integers. Both can be either \latex{ 1 }, \latex{ 2 }or \latex{ 3 }. Thus n is either \latex{ 3 }, \latex{ 4 } or \latex{ 5 }. This means that the faces of a regular solid can only be triangles, squares or pentagons. The number of edges can be expressed from the above equations:
\latex{e=\frac{2kn}{2k+2n-kn}}.
The possibilities are summarized in the following table.
\latex{ n }
tetrahedron
\latex{ k }
\latex{ e }
\latex{ v }
\latex{ f }
octahedron
icosahedron
hexahedron
dodecahedron
\latex{ 3 }
\latex{ 3 }
\latex{ 6 }
\latex{ 4 }
\latex{ 4 }
\latex{ 3 }
\latex{ 4 }
\latex{ 12 }
\latex{ 6 }
\latex{ 8 }
\latex{ 3 }
\latex{ 5 }
\latex{ 30 }
\latex{ 12 }
\latex{ 20 }
\latex{ 6 }
\latex{ 8 }
\latex{ 12 }
\latex{ 3 }
\latex{ 4 }
\latex{ 5 }
\latex{ 3 }
\latex{ 30 }
\latex{ 20 }
\latex{ 12 }
An interesting issue about polyhedra is the \latex{ 7 }-faced polyhedron whose arbitrary two faces are adjacent, constructed by Hungarian mathematician Lajos Szilassi from Szeged.
For each set of data there exists a regular solid. This also means that there exists only \latex{ 5 } regular solids. Their names come from the ancient Greek numbers, thus the solid with four faces is called tetrahedron. (Figure 25)
tetrahedron
octahedron
icosahedron
hexahedron (cube)
dodecahedron
Figure 25
One can meet regular solids not only in technology and architecture, but also in nature: these can be found in crystal structures.
It is worth noting that every regular solid has a circumscribed sphere and there exists a sphere for any one of them which meets every face in a point.
Example 1
Determine the radius of the inscribed and circumscribed spheres of the tetrahedron with sides of length \latex{ a }.
Solution
Generally speaking, in problems about the inscribed and circumscribed spheres it is worth drawing the corresponding planar sections.
In this specific problem, many things follow from the symmetry of the regular tetrahedron. (Figure 26)
- The circumcentre and the incentre coincide.
- The inscribed sphere meet the faces in their centroids.
- \latex{m = r + R}, where \latex{ m } is the altitude of the tetrahedron, \latex{ r } and \latex{ R } are the radius of the inscribed and circumscribed circle, respectively.
\latex{ r }
\latex{ R }
\latex{ R }
\latex{ R }
\latex{ R }
\latex{ R }
\latex{ r }
Figure 26
Intersect the tetrahedron by a plane which contains one edge and the centre of the spheres. This way we obtain the isosceles triangle \latex{ CDF } seen in Figure 27, whose legs are the medians of the faces.
In this triangle, the altitude corresponding to one leg equals the altitude of the tetrahedron. The foot of the altitude is the centroid of the regular triangle, therefore point T is the trisecting point of the segment \latex{ FC }:
\latex{TC=\frac 23\times FC=\frac 23\times\frac{\sqrt3}{2}\times a=\frac{\sqrt3}{3}\times a}.
Applying the Pythagorean theorem for the right angled triangle \latex{ DTC } yields
\latex{a^2=m^2+TC^2},
from which
\latex{m=\sqrt{a^2-\left(\frac{\sqrt3}{3}\times a\right)^2}=\frac{\sqrt6}{3}\times a}.
\latex{ U }
\latex{ r }
\latex{ r }
\latex{ T }
\latex{ K }
\latex{ R }
\latex{ R }
\latex{ C }
\latex{ B }
\latex{ F }
\latex{ A }
\latex{ D }
Figure 27
Now consider the triangles \latex{ DUK } and \latex{ DTF }.
These are similar, since two of their angles are equal.
These are similar, since two of their angles are equal.
By the ratio of the corresponding sides,
\latex{\frac{r}{\frac{\sqrt3}{3}\times a}=\frac{\frac{\sqrt3}{6}\times a}{\frac{\sqrt6}{3}\times a}}.
After rearranging and simplifying,
\latex{r=\frac{\sqrt6}{12}\times a}
We can use the expression connecting the altitude and the radii:
\latex{R=m-r=\frac{\sqrt6}{3}\times a-\frac{\sqrt6}{12}\times a=\frac{\sqrt6}{4}\times a}.
It is easy to see from these results that \latex{\frac R r=3}, which means that the centre of the spheres divides the altitude by a ratio of \latex{ 1 : 3 }.
Exercises
{{exercise_number}}. Is it possible to construct a solid different from the cube with all faces being squares?
{{exercise_number}}. A few identical cubes are placed on the table. The figure shows what can be seen from the front and from the side.
How many cubes are there on the table at most and at least?
front view
side view
{{exercise_number}}. Draw the Schlegel diagram of the five regular solids.
{{exercise_number}}. Determine the radius of the circumscribed and inscribed sphere of the cube with edges of length \latex{ a }. Find the radius of the sphere which is tangent to the lines determined by the edges of the cube.
{{exercise_number}}. The radius of the circumscribed sphere of a regular tetrahedron is \latex{ 10\, cm } longer than that of the inscribed sphere. Determine the length of the edges of the tetrahedron.
{{exercise_number}}. We inscribe a square prism, with length of the edges of its base being half of the other edges, in a sphere with radius of \latex{ 10\, cm }. Determine the length of the edges of the prism.
{{exercise_number}}. On one edge of a cube with edges of length a we pick a point \latex{ P } different from the vertices. Determine the shortest closed path starting from \latex{ P }, traversing every face of the cube and ending in \latex{ P }. How long is this minimal path?
{{exercise_number}}. Find the radius of the inscribed sphere of the regular tetrahedron which is determined by four pairwise non-adjacent vertices of a cube with edges of length \latex{ a }.
{{exercise_number}}. Prove that every convex polyhedron has a face with at most \latex{ 5 } vertices.
Puzzle
Is there any solids with front, side and plan views being as seen in the figure?
side view
front view
plan view