Download Platonic Geogami

Platonic Geogami
By
Katie Snyder: [email protected]
Debra Zibreg: [email protected]
Michelle Dynarski: [email protected] Connie Hadley: [email protected]
Focus: Applications of Platonic Solids
Introduction: Building is fun! Platonic solids created from origami can be used in both the 6th grade
and Geometry curricula. In the 6th grade curriculum, students need to learn probability and Ancient
History, and both can be learned through the use of Platonic solids. These solids are also a fun and useful
tool for students in Geometry to learn about Euler’s Formula, surface area, volume, and interior and
exterior angles of polyhedra.
NYS Math Standards:
6.S.9 List possible outcomes for compound events.
6.S.10 Determine the probability of dependent events.
6.S.11 Determine the number of possible outcomes for a compound event by using the fundamental
counting principle and use this to determine the probabilities of events when the outcomes have equal
probability.
A.G.1 Find the area and/or perimeter of figures composed of polygons and circles or sectors of a
circle. Note: Figures may include triangles, rectangles, squares, parallelograms, rhombuses,
trapezoids, circles, semi-circles, quarter-circles, and regular polygons (perimeter only).
G.PS.2 Observe and explain patterns to formulate generalizations and conjectures.
G.CM.2 Use mathematical representations to communicate with appropriate accuracy, including
numerical tables, formulas, functions, equations, charts, graphs, and diagrams.
G.CN.8 Develop an appreciation for the historical development of mathematics.
G.R.1 Use physical objects, diagrams, charts, tables, graphs, symbols, equations, or objects created
using technology as representations of mathematical concepts.
G.G.36 Investigate, justify, and apply theorems about the sum of the measures of the interior and
exterior angles of polygons.
G.G.37 Investigate, justify, and apply theorems about each interior and exterior angle measure of
regular polygons.
Objectives: Following the conclusion of this lesson, students should be able to:
 understand and apply concepts of probability;
 use visualization and spatial reasoning to analyze characteristics and properties of geometric shapes;
 identify and justify geometric relationships, formally and informally.
Materials Needed: Origami paper, directions, writing utensil
Classroom Applications:
1.
History of Platonic solids.
2. Construct Platonic solids using
origami.
3.
Labeling Platonic solids.
4. Differentiate between prisms and
Platonic solids.
5.
Euler’s Formula for convex polyhedra.
6. Surface area and volume
7. Interior and exterior angles
8. Probability
A Brief History of Platonic Solids
Figure 1
Tetrahedron
Figure 2 Cube
or
Hexahedron
Figure 3
Octahedron

An Etruscan dodecahedron, dated at 500 BCE, was
found in Padova, Italy, and Neolithic stones have been
found in Scotland that resembles Platonic solids
believed to date around 2000 B.C.

The Pythagoreans (6th century B.C.) are known to
have considered the solids in their quest in
simplifying the world around them to numbers.

Plato, from whom the solids received their name,
thought the platonic solids made up the universe. It is
believed that he developed this idea from the Pythagoreans, but this has not been
proven with any solid evidence. He documented in his book The Timaeus (355 B.C.E)
a conjecture that the earth was composed of cubes, air was composed of octahedrons,
water was composed of icosahedrons, and fire was made of tetrahedrons. He also
presumed that dodecahedrons made up the cosmos. Some believe this is because it has
12 faces which correspond to the zodiac. Others think that the golden ratio found in the
pentagon face of the dodecahedron is the reason Plato associated it with the cosmos.

Around 200 BCE, in Book 13 of Euclid’s Elements, Euclid proved that there can only
be five Platonic solids.

In 1597 Johannes Kepler published The
Cosmographic Mystery, stating that the distances
of the planets’ orbits were related to the Platonic
solids. In the year 1600, Kepler was invited by
Tycho Brahe to become his assistant in Prague.
Upon reviewing Tycho’s astronomical
observations, Kepler concluded that the orbits
were elliptical instead, a view that we know to be
true today.

Around 1750 Euler discovered that if a
polyhedron is topologically a sphere, then the
number of vertices plus the number of faces
minus the number of edges equals two. In 1794,
Legendre proved this, giving us more information
Figure 4
Dodecahedron
Figure 5
Icosahedron
Figure 6 Etruscan Dodecahedron:
Emmer, Michele, Ed. The Visual Mind:
Art and Mathematics. Cambridge: MIT
Press, 1993. p216
Figure 7 Kepler's Plantonic solid model of the solar
system from Mysterium Cosmographicum (1596)
about the properties of regular polyhedra.
Differentiating between Platonic Solids and Prisms
The Five Platonic Solids - What makes them so special?
These are the only possible regular polyhedra. They are all regular solids. The same number of
sides meet at the same angles at each vertex, and they have identical polygons that meet at the
same angles at each edge. Each Platonic solid is also identified as a convex polyhedron because
each face lies in a plane that does not intersect the rest of the polyhedron.
Cube
Tetrahedron
#
Vertices
#
Edges
Octahedron
#
Faces
Dodecahedron
# of edges at
each vertex
n-gon for
sides
Icosahedron
# of faces and face
shape name
Tetrahedron
Cube
Octahedron
Dodecahedron
Icosahedron
Prisms: A prism is formed with two parallel congruent polygons (the bases - top and bottom)
that are connected at the edges (the bases - top and bottom) with rectangles.
Triangular
Prism
Rectangular
Prism
Pentagonal
Prism
#
#
Vertices Edges
Triangular Prism
Rectangular
Prism
Pentagonal Prism
Hexagonal Prism
Hexagonal Octagonal Decagonal Dodecagonal
Prism
Prism
Prism
Prism
# of edges at n-gon for # of faces and face
#
each vertex
sides
shape name
Faces
Teachers Copy: Differentiating between Platonic Solids and Prisms
The Five Platonic Solids - What makes them so special?
These are the only possible regular polyhedra. They are all regular solids. The same number of
sides meet at the same angles at each vertex and they have identical polygons that meet at the
same angles at each edge. A Platonic solid is also identified as a convex polyhedron because
each face lies in a plane that does not intersect the rest of the polyhedron.
Hexahedron
Tetrahedron
Dodecahedron
Octahedron
Icosahedron
#
Vertices
#
Edges
#
Faces
# of edges at
each vertex
n-gon for
sides
# of faces and face
shape name
Tetrahedron
4
6
4
3
3-gon
4 triangular
Cube
8
12
6
3
4-gon
6 square
Octahedron
6
12
8
4
3-gon
8 triangular
Dodecahedron
20
30
12
3
5-gon
12 pentagonal
Icosahedron
12
30
20
5
3-gon
20 triangular
Prisms: A prism is formed with two parallel congruent polygons (the bases - top and bottom)
that are connected at the edges (the bases - top and bottom) with rectangles.
Triangular
Prism
Rectangular
Prism
Pentagonal
Prism
#
#
Vertices Edges
Triangular Prism
Rectangular
Prism
Pentagonal Prism
Hexagonal Prism
Hexagonal Octagonal Decagonal Dodecagonal
Prism
Prism
Prism
Prism
# of edges at n-gon for
# of rectangular
#
each vertex
base
faces
Faces
6
9
5
3
3-gon
3
8
12
6
3
4-gon
4
10
15
7
3
5-gon
5
12
18
8
3
6-gon
6
Guided Practice: Convex Polyhedra & Euler’s Formula
Key Terms:
Convex Polygon/Polyhedron: A polygon/polyhedron is convex if every line segment connecting any
two points on or within the polygon/polyhedron lies entirely in the polygon’s/polyhedron’s interior.
Any polyhedron that is topology a sphere will satisfy Euler’s formula.
Euler’s Formula: No. of Vertices – No. of Edges + No. of Faces = 2.
Example: A tetrahedron has four vertices, four faces, and six edges. Substituting the numbers in we have:
4-6+4=2
Therefore a tetrahedron satisfies Euler’s formula.
Verifying a Polyhedron Satisfies Euler’s Formula:
1) Figure out how many faces (F), edges (E), and vertices (V) the polyhedron has.
2) Write down Euler’s Formula:
V–E+F=2
3) Substitute for V, E, and F in the formula.
4) If the left side is equal to 2, then the polyhedron satisfies Euler’s formula.
Exercise:
Fill in the chart below to verify that the Platonic solids satisfy Euler’s formula.
Name:
Number of
Vertices
Number of
Faces
Number of
Edges
Tetrahedron
Cube
Octahedron
Dodecahedron
Icosahedron
Do all the platonic solids satisfy Euler’s formula?
Note: Not all polyhedra that satisfy Euler’s formula are convex!
V–E+F =
Euler’s
Formula
Satisfied?
Guided Practice: Convex Polyhedra & Euler’s Formula
(Teacher’s Copy)
Key Terms:
Convex Polygon/Polyhedron: A polygon/polyhedron is convex if every line segment connecting any
two points on or within the polygon/polyhedron lies entirely in the polygon’s/polyhedron’s interior.
Any polyhedron that is topology a sphere will satisfy Euler’s formula.
Euler’s Formula: No. of Vertices – No. of Edges + No. of Faces = 2.
Example: A tetrahedron has four vertices, four faces, and six edges. Substituting the numbers in we have:
4-6+4=2
Therefore a tetrahedron satisfies Euler’s formula.
Verifying a Polyhedron Satisfies Euler’s Formula:
1) Figure out how many faces (F), edges (E), and vertices (V) the polyhedron has.
2) Write down Euler’s Formula:
V–E+F=2
3) Substitute for V, E, and F in the formula.
4) If the left side is equal to 2, then the polyhedron satisfies Euler’s formula.
Exercise:
Fill in the chart below to verify that the Platonic solids satisfy Euler’s formula.
Name:
Number of
Vertices
Number of
Faces
Number of
Edges
V–E+F =
Tetrahedron
Cube
Octahedron
Dodecahedron
Icosahedron
4
8
6
20
12
4
6
8
12
20
6
12
12
30
30
2
2
2
2
2
Do all the Platonic solids satisfy Euler’s formula? Yes
Note: Not every polyhedron that satisfies Euler’s formula is convex.
Euler’s
Formula
Satisfied?
Yes
Yes
Yes
Yes
Yes
Guided Practice: Interior & Exterior Angles of Regular Polygons
Key Terms:
Interior Angles:
To find the total measure of all interior angles in a polygon:
1) Find the number of sides. Let this be n.
2) Plug n into interior angle formula:
(n – 2) x 180
3) To find the measure of each interior angle in a regular polygon, divide your answer to number
2 by the number of sides (n).
Exterior Angles:
To find the total measure of each exterior angle in a regular polygon:
1) Find the number of sides. Let this be n.
2) Plug into the exterior angle formula:
Regular
Polygon
Face of an
Icosahedron
Octagon
Face of a
Dodecahedron
Hexagon
Face of a
Cube
Nonagon
Number of Sides
(n)
Measure of Each
Interior Angle
Measure of Each
Exterior Angle
Guided Practice: Interior & Exterior Angles of Regular Polygons
(Teacher’s Copy)
Key Terms:
Interior Angles:
To find the total measure of all interior angles in a polygon:
1) Find the number of sides. Let this be n.
2) Plug n into interior angle formula:
(n – 2) x 180
3) To find the measure of each angle in a regular polygon, divide your answer to number 2 by the
number of sides (n)
Exterior Angles:
To find the measure of each exterior angle in a regular polygon:
1) Find the number of sides. Let this be n.
2) Plug into exterior angle formula:
Regular
Polygon
Face of an
Icosahedron
Number of Sides
(n)
Measure of Each
Interior Angle
Measure of Each
Exterior Angle
3
60
120
Octagon
8
135
45
Face of a
Dodecahedron
5
108
72
Hexagon
6
120
60
Face of a
Cube
4
90
90
Nonagon
9
140
40
Finding Surface Area & Volume of Platonic Solids
Key Terms:
Area of a Polygon: The unique real number assigned to any polygon which indicates the
number of non-overlapping square units contained in the polygon’s interior.
Surface Area: The sum of the areas of all the faces or curved surfaces of a solid figure.
Inradius ( ): For a regular solid, the radius of its inscribed sphere. For each Platonic solid,
there is a formula for the inradius in terms of the side length – see table below.
Volume: A measure of the number of cubic units needed to fill the space inside a solid figure.
-Formula to find the volume of a regular solid:
V = f ( )(face area)(
),
where f is the number of faces and face area is the area of any face of the regular polyhedron.
Fill in the table below:
Platonic
Solid:
Tetrahedron
Number
of
Faces:
Length
Face
of
Shape: Base
(cm)
Length
of
Inradius
(cm)
1
0.20412
1
0.5
Octahedron
1
0.40825
Dodecahedron
1
1.11352
Icosahedron
1
0.75576
Cube
Area
of
One
Face:
(cm2)
Surface
Area:
(cm2)
Volume
of
Figure:
(cm3)
Challenge Question: Explain how to find the Area of a face in the Dodecahedron.
Finding Surface Area & Volume of Platonic Solids
(Teacher’s Copy)
Key Terms:
Area of a Polygon: The unique real number assigned to any polygon which indicates the
number of non-overlapping square units contained in the polygon’s interior.
Surface Area: The sum of the areas of all the faces or curved surfaces of a solid figure.
Inradius ( ): For a regular solid, the radius of its inscribed sphere.
Volume: A measure of the number of cubic units needed to fill the space inside a solid figure.
-Formula to find the volume of a regular solid:
V = f ( )(face area)(
),
where f is the number of faces and face area is the area of any face of the regular polyhedron.
Fill in the table below:
Length
of
Inradius:
Side:
(cm)
(cm)
Area
of
One
Face:
(cm2)
Surface
Area:
(cm2)
Volume:
(cm3)
Platonic Solid:
Number
of
Faces:
Face
Shape:
Tetrahedron
4
Triangle
1
0.20412
√3/4
√3
√2/12
Cube
6
Square
1
0.5
1
6
1
Octahedron
8
Triangle
1
0.40825
√3/4
2√3
√2/3
Dodecahedron
12
Pentagon
1
1.11352
1.720
20.646
7.663
Icosahedron
20
Triangle
1
0.75576
√3/4
5√3
2.182
Challenge Question: Explain how to find the Area of a face in the Dodecahedron.
A pentagon consists of 5 triangles. Find the area of one triangle, multiply by 5.
Name______________________________
Guided Practice: Platonic Solids and Probability
Each pair of students has a collection of polyhedra consisting of the five origami Platonic solids. Fill in
the chart below for the number of faces of each color that each Platonic Solid has. Your total in a given
column should equal the number of faces for that Platonic solid.
Tetrahedron
Cube
Octahedron
Dodecahedron
Icosahedron
# Red Faces:
# Blue Faces:
# Yellow Faces:
Total:
Based on the above chart, calculate the probability of rolling each color. When you add the probabilities
in any given column, they should sum to one.
Tetrahedron
Cube
Octahedron
Dodecahedron
Icosahedron
Probability of
Rolling Red
Probability of
Rolling Blue
Probability of
Rolling Yellow
Total:
1 – 4 Circle the event that is more likely to happen. Hint: Use the information that you calculated above.
If the likelihood of the two events is the same, write “Equal.”
1. Tetrahedron Landing on Yellow OR Octahedron Landing on Blue
2. Cube Landing on Blue OR Dodecahedron Landing on Red
3. Tetrahedron Landing on Red OR Icosahedron Landing on Yellow
4. Icosahedron Landing on Blue OR Dodecahedron Landing on Red
5 – 8 If we roll two of the Platonic solids, what is the probability that exactly one of them will land on
RED?
5. If we roll a Tetrahedron and an Octahedron:
______
□ ______ = ______
□ ______ = ______
If we roll a Dodecahedron and a Cube: ______ □ ______ = ______
If we roll a Tetrahedron and an Icosahedron: ______ □ ______ = ______
6. If we roll a Cube and an Icosahedron: ______
7.
8.
9 – 12 If we roll two of the Platonic solids, what is the probability that both of them will land on BLUE?
9. If we roll a Tetrahedron and an Octahedron: ______
□ ______ = ______
□ ______ = ______
11. If we roll a Dodecahedron and a Cube: ______ □ ______ = ______
12. If we roll a Tetrahedron and an Icosahedron: ______ □ ______ = ______
10. If we roll a Cube and an Icosahedron: ______
Roll each Platonic solid and keep track of the number of times you land on each color (Tetrahedron: Roll
16 times, Cube: Roll 24 times, Octahedron: Roll 32 times, Dodecahedron: Roll 50 times, Icosahedron:
Roll 80 times)
Enter the number of times you land on each color over the total number of rolls for that solid.
Your total for any given solid should be ONE:
Tetrahedron
Red Faces:
Blue Faces:
Yellow Faces:
Total:
Cube
Octahedron
Dodecahedron
Icosahedron
Review your results. Use some specific comparisons of sample frequencies vs. theoretical probabilities.
Be sure to list some reasons why the sample frequencies and theoretical probabilities might differ.
Guided Practice: Platonic Solids and Probability- Teacher’s Copy
Each pair of students has a collection of polyhedra consisting of the five origami Platonic solids. Fill in
the chart below for the quantity of each color that each Platonic solid has. Your total should equal the
number of faces each Platonic solid has.
Tetrahedron
Cube
Octahedron
Dodecahedron
Icosahedron
# Red Faces:
2
1
3
4
7
# Blue Faces:
1
2
4
4
5
# Yellow Faces:
1
3
1
4
8
Total:
4
6
8
12
20
Based on the above chart, calculate the probability of rolling each color. When you add the probabilities
for the colors they will add to one.
Tetrahedron
Cube
Octahedron
Dodecahedron
Icosahedron
Probability of
Rolling Red
1/2
1/6
3/8
1/3
7/20
Probability of
Rolling Blue
1/4
1/3
1/2
1/3
¼
Probability of
Rolling Yellow
1/4
1/2
1/8
1/3
2/5
1
1
1
1
1
Total:
1 – 4 Circle the event that is more likely to happen. If the likelihood of the two events is the same, write
“Equal”.
1. Tetrahedron Landing on Yellow OR Octahedron Landing on Blue
2. Cube Landing on Blue OR Dodecahedron Landing on Red - Equal
3. Tetrahedron Landing on Red OR Icosahedron Landing on Yellow
4. Icosahedron Landing on Blue OR Dodecahedron Landing on Red
5 – 8 If we roll two of the Platonic solids, what is the probability that exactly one of them will land on
RED?
5.
6.
7.
8.
If we roll a Tetrahedron and an Octahedron: (1/2)(5/8) + (1/2)(3/8) = 1/2
If we roll a Cube and an Icosahedron: (1/6)(13/20) + (5/6)(7/20) = 48/120 = 2/5
If we roll a Dodecahedron and a Cube: (1/3)(5/6) + (2/3)(1/6) = 7/18
If we roll a Tetrahedron and an Icosahedron: (1/2)(13/20) + (1/2)(7/20) = 1/2
9 – 12 If we roll two of the Platonic solids, what is the probability that both of them will land on BLUE?
9.
10.
11.
12.
If we roll a Tetrahedron and an Octahedron: 1/4 × 1/2 = 1/8
If we roll a Cube and an Icosahedron: 1/3 × 1/4 = 1/12
If we roll a Dodecahedron and a Cube: 1/3 × 1/3 = 1/9
If we roll a Tetrahedron and an Icosahedron: 1/4 × 1/2 = 1/8
Roll each Platonic solid and keep track of the number of times you land on each color (Tetrahedron: Roll
16 times, Cube: Roll 24 times, Octahedron: Roll 32 times, Dodecahedron: Roll 50 times, Icosahedron:
Roll 80 times)
Enter the number of times you land on each color over the total number of rolls for that solid.
Your total for any given solid should be ONE:
Tetrahedron
Cube
Octahedron
Dodecahedron
Icosahedron
Red Faces:
Blue Faces:
Yellow Faces:
Total:
Review your results. Use some specific comparisons of sample frequencies vs. theoretical probabilities.
Be sure to list some reasons why the sample frequencies and theoretical probabilities might differ.
DO THE POLYHEDRON
Everybody’s doing a brand new shape now
Come on baby Do the Polyhedron
I know you get to roll it to find your chance now
Come on baby do the polyhedron
With origami you can do it with ease
It’s easier than learning your Kappa Delta Phi’s
So come on come on
and do this origami with me
You’ve gotta crease your edge now
Come on baby fold up fold back
Well I think you’ve got the knack
Now that you can do it let’s make a solid now
Come on baby do the polyhedron
A fold it, mold it motion like a Dr. Straight now
Come on baby do the polyhedron
Do it nice and straight now don’t lose control
A little bit of algebra and a lot of folds
Come on come on do the polyhedron with me
Yeah yeah yeah yeah
Move around your paper to make your creation
(Come on baby, do the polyhedron)
Do it Makin’ Solids like you got the notion,
(Come on baby, do the polyhedron)
There's never been an angle that’s so easy to see,
It even makes math history seem like a breeze)
So come on, come on and,
do the polyhedron with me