The Five Platonic Solids (Origami activity)

The Five Platonic Solids (Origami activity)
Elementary and High school Math
Tetrahedron
Hexahedron or Cube
Octahedron
Icosahedron
Dodecahedron
There are infinitely many types of polyhedron, but there are only five regular polyhedrons –
called Platonic solids. They are named after Plato, who discovered them independently about
400 B.C. The Pythagoreans knew their existence before this time, and Egyptians had used
some of them in architecture and other objects they designed.
<Definition>
Platonic solids are convex solids whose edges form congruent regular plane polygons.
※Plato proved that there were only five possible regular convex solids.
Let’s make Platonic solids by folding paper! (good for 4 lessons)
1st lesson: To acquaint with the history of Platonic solids and be able to make Hexahedron
(Cube) by folding paper.
2nd lesson: To be able to make an equilateral triangle by folding paper and assemble
Tetrahedron, Octahedron and Icosahedron.
3rd lesson: To be able to make a pentagon by folding paper and assemble Dodecahedron.
4th lesson: To summarize characteristics of each Platonic solid
1st lesson:
● Preparation
1 A sheet of paper (short, A4, long, news paper etc.) for each student.
○
2
○
Models of Platonic solids or Manila paper which Platonic solids are drawn
●Procedure
1
○
Show the model of Platonic solids to students and tell the history and the concept of them.
(Referring to the previous page)
2 Instruct students to form Cube by folding paper.
○
Q: Which solid is the most familiar with you among Platonic solids?
2 - a: Q: Do you know how to make a square from a rectangle?
○
rectangle
Cube
square
※ Since the length of width is equal to the length of height, we can form a square.
2 – b: We are going to make a cube from this square by only folding the paper.
○
Fold and open to make
a diagonal line
Fold again
Do the same with
the other side
Now, we have a square.
From the vertex on the base fold the paper to the
top (Do the same with the other vertices)
From the side, fold the
paper to the opposite side
Open both surfaces of the square
to be able to form triangles
From the sides fold the
paper to the center
Do the same with the other
side (the back portion as well)
From the top, fold the paper of the
pocket to the center. Fold again along
the line and insert it
Fold and open to
make creases
Blow and make
creases tightly
D
A
H
3 Name each vertex and check the characteristics of a cube
○
B
E
<Sample questions>
Question
1
2
3
4
5
6
7
8
9
10
11
12
13
C
F
G
Answer
How many surfaces are there?
6
Do you know another name of a Hexahedron, because the
cube?
number of surfaces is 6.
How many sides are there?
12
How many vertices are there?
8
What is the shape of surfaces?
Square
Which is longer, EG or BD ? Equal
Which surfaces are
□ AEFB, □ HFGC, □ CGHD
perpendicular to□ ABCD?
□ DHEA
Which surfaces are parallel to
□ CGHD
□ AEFB?
How many surfaces are
4
perpendicular to □ HFGC?
How many sides
4
intersect FG ?
List up line segments which is
AB, AD, EH , EF
perpendicular to AE .
Which line segments are
AE , BF , CG
parallel to DH ?
What is the name of figure
ABGH?
Rectangle
4 Remind students to keep a cube in good condition and bring glue for the next lesson
○
2nd lesson:
● Preparation
1
○
Three sheets of paper (short, A4, long, news paper etc.) for each student.
2
○
The nets for three Platonic solids that are drawn on the Manila paper
●Procedure
1 Review 5 Platonic solids
○
2 Instruct students to form an equilateral triangle by folding paper.
○
Q: Some of the Platonic solids have common characteristics with each other. Have you
notice that?
The most visible common characteristic is the shape of surface.
The common shape of surfaces is
Tetrahedron
an equilateral triangle among
three.
Octahedron
Icosahedron
2 - a : We are going to prepare a square first. We need ¼ size of a square.
○
rectangle
square
Divide and cut into 4
2 - b: We are going to make an equilateral triangle from a square by paper folding.
○
¼ size of a
square
Fold and open to make a crease
Do the same from the
vertex on the left
Fold both parts
From the vertex, fold the
paper to the crease to be able
to form a right angle triangle
Fold the paper to mark the
midpoint of the upper side
Turn it over
Fold the paper from the bottom to
the midpoint of the upper side
Fold the paper along the
crease from the left vertex
Fold the small part inward
Fold the paper along the
crease from the right vertex
This equilateral triangle has pockets along all sides!!
And insert it
2
- c: It seems that we will be able to create these three Platonic solids using this triangle.
○
But, we need connections.
¼ size of
a square
Divide into 4 and cut
1/16 size of
a square
Fold and open to make a crease
From the corners fold
the paper to the center
Equilateral triangles
You can connect two
equilateral triangles
with this connection.
2 - d: Make groups. Each groups has three members.
○
We are going to form Tetrahedron, Octahedron and Icosahedron with members of
the group.
Q: How many equilateral triangles do you need to form these three solids?
4(Tetrahedron)+8(Octahedron)+20(Icosahedron)=32 pieces
Q: How about connects?
4 × 3 ÷ 2+8 × 3 ÷ 2+20 × 3 ÷ 2=96 ÷2= 48 pieces
※ Students might not be able to get the exact number of connections. If not,
just let them proceed. Teachers may check after students finished forming
☆ Shall we start making them?
● Prepare the glue. Then, you may start making three Platonic solids by
connecting equilateral triangles.
Let students try to make three Platonic solids by trial and error. The only hint teacher is
going to show is the models or sketch of them. We should give priority to the process of
making. Students may get the sense of 3-Dimention through making them.
(Sample of Nets)
※If students encounter the difficulty of making, you
may show these nets as a hint.
Tetrahedron
Octahedron
Icosahedron
3
○
Remind students to keep three Platonic solids they made in a good condition and
bring glue to the next lesson
3rd lesson:
● Preparation
1 Three sheets of paper (short, A4, long, news paper etc.)
○
for each student.
2
○
The net for Dodecahedron which is drawn on the Manila paper
●Procedure
1
○
Review 5 Platonic solids again
Dodecahedron
Q: What Platonic solid haven’ we made yet?
Dodecahedron
2 Instruct students to form a regular pentagon by folding paper.
○
2 - a: We are going to make a pentagon from a rectangle by folding.
○
Divide and cut into 8 pieces
1/8
Tie the paper as well as you tie a rope
From the left side, fold the
paper along the base
From the right side, fold the
paper along the upper right side
Fold and insert it
Fold and insert it
Cut the lower right side
This regular Pentagon
has pockets on each
side!
Insert the both parts to be able
to create regular Pentagon
2 - b : It seems that we will be able to create dodecahedron using this pentagon. But, we
○
need connections.
1/8
Fold the paper to be able to create a square
Fold it again
Fold it along the line
Open it
Make it double
Cut it along the line
Connection
Fold and open it to make a crease
You can connect two pentagons.
● Prepare the glue. Then, you may start making a Dodecahedron by connecting
pentagons.
As well as the other Platonic solids, let students try to make Dodecahedron by trial and
error.
※If students encounter the difficulty of making, you
may show the net as a hint.
We need;
12 regular pentagons
30 connections
Dodecahedron
3 Remind students to keep Dodecahedron in a good condition and bring all Platonic solids
○
that they made to the next lesson
4th lesson:
● Preparation
1 Five Platonic solids that students made in previous lessons
○
●Procedure
1
○
Review 5 Platonic solids again
Q: What is the definition of the Platonic solid?
Platonic solids are convex solids whose edges form congruent regular plane polygons.
Q: Check Platonic solids that you made in terms of the definition.
<Activity sheet>
Name of solid
Tetrahedron
Hexahedron
Octahedron
Dodecahedron
Icosahedron
NO. of
surfaces
Figure on the
edges
(blank)
(blank)
(blank)
(blank)
Q: We are going to look into these Platonic solids in detail. Aside from the definition,
from what perspective can we check the characteristics of these solids? Write down
the perspective in the blank on the activity sheet and start to check them
respectively.
(EX) No. of sides, No. of vertices, No. of surfaces that are gathering around one vertex etc…
<Activity sheet> (EXAMPLE)
Characteristics
Tetrahedron
Hexahedron
Octahedron
Dodecahedron
NO. of surfaces
Figure on the
edges
4
Triangle
6
Square
8
Triangle
12
Pentagon
20
Triangle
Icosahedron
No. of surfaces that
are gathering around
each vertex
3
3
4
3
5
No. of vertices
4
8
6
20
12
No. of sides
6
12
12
30
30
★ Another common characteristic is the No. of surfaces that are gathering around each
vertex.
Q: Have you notice that all Platonic solids have a certain relationship among No. of
surfaces, No. of vertices and No. of sides.
(If students can’t notice, you may give them a hint)
<HINT1> Add No. of surfaces and No. of vertices.
<HINT2> Compare the sum of two numbers and No. of sides
Q: Can you make the formula for these numbers?
No. of surfaces + No. of vertices – 2 = No. of sides
This formula is known as a Euler’s theory
☆ Definition:
Platonic solids are convex solids whose edges form congruent regular plane
polygons.
(※ Only 5 Platonic solids exist)
★ Common characteristics:
1
○
The number of gathering surfaces at each vertex is constant.
2
○
Euler’s theory
No. of surfaces + No. of vertices – 2 = No. of sides
Ateneo De Davao University – Regional Science Teaching Center
<Activity sheet (Platonic Solids)>
Group name:
Characteristics
Tetrahedron
Hexahedron
Octahedron
NO. of
surfaces
Figure on
the edges
4
6
8
12
20
Triangle
Square
Triangle
Pentagon
Triangle
<Activity sheet (Platonic Solids)>
Characteristics
NO. of
surfaces
Figure on the
edges
Tetrahedron
Dodecahedron
Icosahedron
Group name:
Hexahedron
Octahedron
Dodecahedron
Icosahedron
4
6
8
12
20
Triangle
Square
Triangle
Pentagon
Triangle