1. Platonic Solids: are convex polyhedrons whose faces are regular polygons equal and have
the same number of faces that are located at each vertex.
There are only five platonic solids, which are as follows:
- Association of the Platonic solids the elements of nature:
The five elements
Tetrahedron (Fire):
Formed by four faces, equilateral triangles, and each vertex competes
three sides. The prefix comes from the Greek tetra, meaning four (four
sides). According with Plato, this solid is fire because the atom of fire
has a polyhedron shape with four sides (tetrahedron).
Cube (Earth):
The cube is the only regular polyhedron with square faces. The cube
has six faces, so you can also call hexahedron (hesa means six in
Greek). This represents the solid earth, because Plato believed and
asserted that the atoms of the earth would be cubes which allowed to
be placed perfectly side by side, giving them strength.
Octahedron (Air):
The faces of this polyhedron are also equilateral triangles, but in each
vertex meet four triangles. It consists of eight sides. So that this
polyhedron is called octahedron (octa in Greek means eight). This solid
represent air because according to Plato, an air atom was a
polyhedron with eight sides (octahedron).
Dodecahedron (Cosmos):
The dodecahedron is the only regular polyhedron whose faces are
regular pentagons. It consists of 12 faces regular pentagons and at
each vertex contributes 3 faces. The prefix dodeca meaning twelve in
Greek. This solid represents the universe because according to Plato,
the cosmos would consist of atoms in the shape of dodecahedrons.
Icosahedron (Water):
In this polyhedron are five equilateral triangles that meet at each
vertex, comprising twenty faces. Therefore, the polyhedron is called
icosahedron (Icosa in Greek means 20). This solid represents the
water because according to Plato the water would be constituted by
icosahedrons.
- Check why there are only five platonic solids: Students must use the manipulative polydrons
and build convex polyhedra with faces all the same. How many regular polyhedra can you
construct with equilateral triangles, squares, regular pentagons and regular hexagons?
Equilateral triangles:
As each interior angle is 60 ° at each vertex can be three, four or five triangles.
Three triangles each vertex is obtained a tetrahedron
Four triangles each vertex is obtained an octahedron
Five triangles each vertex is obtained a Icosahedron
Squares:
As each interior angle measures 90 ° can only exist at each vertex 3 squares.
Therefore, there is a cube.
Pentagons:
As each interior angle measures 108 ° at each vertex can only be three. We have the
dodecahedron.
With regular hexagons is not possible because the amplitude of its interior angles is 120
degrees and 3x120 degrees is the same as 360 degrees.
• The teacher should adjust the programmatic content of the school year students, and in
primary education (9 years old) children already speak in Platonic solids (name and definition,
history and its association with the elements of nature, and they will return to study them later.
2. Square wheels:
The secret is in the shape of the road over which the square wheels roll. A square wheel can
roll smoothly, keeping its axle moving in a straight line.
Just as a square rides smoothly across a roadbed of linked inverted catenaries, other regular
polygons, including pentagons and hexagons, also ride smoothly over curves made up of
appropriately selected pieces of inverted catenaries.
A polygon is inscribed in a circle if all its vertices are points on the circle. This circle is said to be
circumscribed to the polygon.
The regular polygons can always be inscribed in a circle, is not the case with non-regular
polygons.
Center of the polygon is the center of the circle circumscribed to it. Radius of the polygon is the
radius of the circle circumscribed to it (r).
Apothem of the polygon is the segment connecting the center point of the polygon with the
medium one of its sides (a).
The difference between the apothem of the polygon (in this case square) and the radius of the
circumcircle gives us the maximum height of the catenaries.
As the number of the polygon’s sides increases, these catenaries segments get shorter and
flatter. Ultimately, for an infinite number of sides (in effect, a circle), the curve becomes a
straight, horizontal line.
Interestingly, triangular wheels don't work. However, you can find roads for wheels shaped like
ellipses, cardioids, rosettes, teardrops, and many other geometric forms.
• The teacher should adapt the mathematical contents discussed here at the age of students:
polygons and circumference.
3. Towers of Brahma, also known as Towers of Hanoi is a game of Oriental origin, consisting of
three pins and some discs on each other in order of decreasing diameter from the bottom up.
The objective is to move the tower of discs from one end peg to the other in a few moves as
possible. Two rules: Move one disc at a time, and never place a larger disc on a smaller one.
Legend: the Tower of Brahma was invented by the French mathematician, Edouard Lucas, in
1883. He was inspired by a legend that tells of a Hindu temple where the pyramid puzzle might
have been used for the mental discipline of young priests. Legend says that at the beginning of
time the priests in the temple were given a stack of 64 gold disks, each one a little smaller than
the one beneath it. Their assignment was to transfer the 64 disks from one of the three poles to
another, with one important provisonal large disk could never be placed on top of a smaller one.
The priests worked very efficiently, day and night. When they finished their work, the myth said,
the temple would crumb.
The number of moves required to solve a Tower of Hanoi puzzle is 2n -1, where n is the number
of disks.[4]
Number of Disks
1
2
3
4
5
Number of Moves
1
3
7
15
31
Powers of two help reveal the pattern:
Number of Disks (n) Number of Moves
1
2^1 - 1 = 2 - 1 = 1
2
2^2 - 1 = 4 - 1 = 3
3
2^3 - 1 = 8 - 1 = 7
4
2^4 - 1 = 16 - 1 = 15
5
2^5 - 1 = 32 - 1 = 31
So the formula for finding the number of steps it takes to transfer n disks from post A to post B
is: 2^n - 1.
From this formula you can see that even if it only takes the monks one second to make each
move, it will be 2^64 - 1 seconds before the world will end. This is 590,000,000,000 years (that's
590 billion years) - far, far longer than some scientists estimate the solar system will last. That's
a really long time!
Concepts covered: Sequences and regularities, successions, progressions geometric, limit of a
function, fractal geometry.
Students in primary schools speak in numerical sequences from 8 years of age. The teacher
should tailor the contents to the age of the students.
4. Turning mirror: With the help of mirrors and investigation sheets (and the module itself)
students will approach concepts such as sum of the internal angles of a triangle, classification of
triangles, regular polygons, bisector and reflection.
Conveniently adjusting with the age of the students, they must conclude, after the activity, that
the bigger the angle formed by the mirrors, smaller the number of sides of the polygon obtained.
Rule: angle x number of sides = 360°
5. Volume of the pyramid: Calculate the formula to the volume of the pyramid (square base)
using the formula to the volume of the cube.
Cube: volume = area of base x height
Pyramid: Volume = 1/3 area of base x height
In the experience of "volume of the pyramid" we have two cases: build the cube using 6 parts or
using only three. At latter case, the teacher could relate this with the fact that volume of the
pyramid is 1/3 the volume of the cube (with the same base and same height). In the case of the
six pyramids, each pyramid has the same base as the cube, but the height is half of the height
of the cube, so 6 pyramids are necessary to obtain the cube.
Teacher should speak in formulas only for students over 10 years of age.