# Here is the sample script and questions first iteration

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This presentation uses an activity from this book:
http://www.amazon.com/Project-Origami-Activities-Exploring-
Mathematics/dp/1568812582

Room set up:
Diagram of “folding business cards” is displayed on a projector screen.

Workshop Intro:

Script:
General instructions:
Our goal is to get the participants to collaborate. We want them to ask questions and learn
something, by the end of the workshop. Each group leader should use pay attention to
their group – are they participating, asking questions, getting stuck, and then decide when
and what questions to ask. Team leaders should gauge their time so that all groups will
have discussed Euler’s (pronounced oiler) Formula before leaving the workshop.

When someone makes a conjecture, respond by saying “I want to believe what you’re

Step 1 Create building blocks:
Now we are going to create the basic building blocks, which are displayed
on the projector.
First let’s create a Left Handed Unit. (Describe how you’re folding, while folding a
We are now creating the left handed unit. Take the lower left corner and fold this to have
it connect with the upper right hand corner. Make sure that your folds are crisp. We want
crisp edges because this will be important later. I will now walk around to make sure that

Next let’s create a Right Handed Unit.

Notice that these simple folds produce equilateral triangles.

Optional discussion topic 1:
Are they really equilateral triangles and how can we tell?
All sides equal, all 60 degree angles

Optional discussion topic 2:
What a polygon is?
 A closed plane figure with N-sides.
 A polygon can be either convex or concave.

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Step 2 Create a pyramid/tetrahedron:
Optional discussion topic 3:
Can you make a pyramid using 1 left hand and 1 right hand unit?

Demonstrate the hugging technique - the small side hugs the larger size. It is free
standing and if you throw it, it should not break apart.

Now we need to think about vertices, edges, and faces.
*On white board, begin following table

Vertices Faces      Edges     Degree of Vertices

The pyramid you just created is called a Tetrahedron.

Optional discussion topic 4:
Anyone know why it’s called a Tetrahedron
Answer: It has four faces/sides and four vertices. Tetra comes from the Greek
word for 4

If someone says at this point”Isn’t this the Euler’s formula?” We say that is where we
are heading. Then you could say, would you be able to explain that to us?

*On white board, update table.

Recurring discussion topic
How many vertices and edges? What is the degree of the vertices?

Vertices     Faces Edges Degree of Vertices
Tetrahedron             4          4     6           3

Optional discussion topic 5:
How many units/biz cards did you need to make a Tetrahedron?

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Step 3 Create an octahedron:
Optional discussion topic 6:
How many left and right units do you need to create an octahedron?

Let’s try to build it.

Optional discussion topic 7:
How many faces does an octahedron have?
(If stuck ask how many faces does the card have?) No card is ever covering a face

*After build, on white board, update table.

Recurring discussion topic
How many vertices and edges? What is the degree of the vertices?

Vertices    Faces Edges Degree of Vertices
Tetrahedron               4         4     6           3
Octahedron                6         8    12           4

Optional discussion topics 8:
Are these convex shapes? Why?

Optional discussion topic 9:
Does anybody know what a convex polygon is? In the 2D planar domain
 Every line segment between two vertices remains inside or on the boundary of the
polygon.
 Every internal angle is less than 180 degrees.

*On white board draw examples of convex and concave polygons. Show the two sample
shapes

In 3D domain
What is vertex?
(Answer: Show on paper that a vertex is where edges come together)

What is degree of vertex? (answer: the number of edges coming into a vertex)

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When talking about 3-d polygons made of 60 degree angle triangles what would be the
maximum number of degree for a vertex that would not be flat? A flat surface in a 3-d
polygon would have what degree? (Answer 360degrees). How many triangles can come
together to in a vertex to make less than 360 degrees if all of the triangles are 60 degrees?

Euler’s Formula:

Discussion topic EF1:
Is anyone familiar with Euler’s Formula?

* On white board write the formula V - E + F = 2

Discussion topic EF2:
Does Euler’s Formula work for the values in our table?

Step 4 Create an icosahedron:
Optional discussion topic 10:
How many faces does an icosahedron have?

Let’s try to build it.

*After build, on white board, update table.

When you were building the icosahedron, did you notice how it would be impossible to
(Answer: because 5 degrees is 300 degrees and 6 degrees would be 360 and we would no
longer be convex) we would be flat.)

Recurring discussion topic
How many vertices and edges? What is the degree of the vertices?

Vertices    Faces Edges Degree of Vertices
Tetrahedron               4         4     6           3
Octahedron                6         8    12           4
Icosahedron              12        20    30           5

Continue as time permits

Optional discussion topic EF3:

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What numbers/values will work? What numbers/values won’t work?

Hints – How many faces does folded business card have? (2)
How many edges does a business card have (3)

What face combinations have we already done?

Basically the shapes that we’ve created with these business cards are called deltahedra.

A deltahedron is a polyhedron whose faces are all equilateral triangles. What other
deltahedra can you build?

Keep in mind that we can use Euler’s Formula to prove we’ve created a deltahedron..

Do they numbers/values have to be even?
(No, we need an even number of face. Can have an odd number of edges and
vertices)
Do the vertices and edges have to be even?
Let’s try to build something with 6 faces

Optional discussion topic EF4:
Could we build something with 22 faces? (We conjectured no, because degree of
vertices would be greater than 5.)

(Only deltahedra with 4,6,8,10,12,14,16,20 will work)

Optional discussion topic EF5:
Could we build something with 18 faces? (We conjectured no, because degree of
vertices would be greater than 5.)

Step 5 partially fill out the table for the participants:

PASS OUT THE ONLY HANDOUT (or show on the projector) – the Johnson
Solids.
The definition of a Johnson solid is a bit recursive and requires a lot of definitions. It isn’t important but here it is in case someone
In geometry, a Johnson solid is a strictly convex polyhedron, each face of which is a regular polygon, but which is not uniform, i.e.,
not a Platonic solid, Archimedean solid, prism or antiprism.

Tell them: So, guess what. There are only 8 convex deltrahedron possible. Do you
believe me?
Let me give you some information and see what you think of it:

Vertices Faces Edges Degree                                  Cards V3 V4 V5 Total
of                                                     V

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Vertices
Tetrahedron          4         4       6        3          2                     12
Octahedron           6         8      12        4          4                     24
Icosahedron         12        20      30        5         10                     60
Triangular           5         6       9                   3     2    3          18
Dipyramid
Pentagonal           7        10      15                  5           5     2    30
Dipyramid
Snub                 8        12      18                  6      0    4     4    36
Disphenoid
Triangmented         9        14      21                  7      0    3     6    42
Triangular
Prism
Dipyramid           10        16      24                  8           2     8    48

Is this data correct?
What convex deltrahedron shapes can we make with these business cards?
What makes it convex?
What numbers of faces are possible?
Can we have odd faces?
Each card adds how many faces? How many vertices? Is it true that each card adds three
edges?
Do you see any patterns in the data?
Are any numbers missing?
Under what conditions can these be built?
Under what conditions are these impossible to be built? (Convex Deltrahedron)
What if we told you that no vertex can have more than 6 degree? Is that true? Why?
If we restrict the number of vertices to 5 at the max – can we use Euler’s formula to
figure out possible shapes?

Optional discussion topic EF6:
What about 18 faces? 22 faces? Why aren’t these possible?

What if you took an Icosahedron and pulled out one card? That would give you 18 faces.
Why wouldn’t you be able to reconnect them?

(Answer: because if you pulled out one card the surrounding vertices would be 4,4,5,5
and it is impossible to reconnect the shape without making a vertex connected with 6.
And this isn’t possible)

Think about how you would go from shape to shape? Can you pull a card out or add a
card from one shape and get another shape?

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(Answers: pull one from snub disphenoid and get pentagonal dipyramid. Remove one
from triaugmented triangular prism and get snub disphenoid. Add two to Gyroelongated
square dipyramid and get an icosohedron)

Think about the combinations of 4 and 5 vertex degree’s that are possible and those that
are not.

Groups can continue to work on creating different shapes.

General instructions:
Thank everyone for attending and participating 

Definitions:
Convex Polygon – Every line segment between two vertices remains inside or on the
boundary of the polygon. And every internal angle is less than 180 degrees.

Deltahedron – a polyhedron whose faces are all equilateral triangles. There are infinitely
many deltahedron, but only 8 of these are convex.

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Johnson Solids – our presentation deals with a subset of Johnson solids, the one with only
triangle faces.

Planar Graph – a graph with non-intersecting edges.
A graph is planar if it can be drawn in a plane without the graph edges crossing. (If any
of the edges intersect/cross then it is not a planar graph).

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Planar Graph examples

Polygon – a closed plane figure with N-sides.

Regular - all vertices have the same degree.

Other interesting questions we considered:
Using the left and right basic building blocks what convex polygons can we build from
these basic units? What can’t you build and why?
Answer: You can build any polyhedron, with equilateral triangle faces and
vertices of 5 or less.

Do you really need two of each type of units make an octahedron?
Answer: No, you can use only the left handed. All the folds will be going exactly
the same way. If you use a left and a right, it may not be as difficult to recreate.

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