Three-dimensional Geometry Investigating Regular Polyhedra– Level 1

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What You’ll Learn…

 What the Platonic solids are, what makes them unique, and how they relate to one another.

 The math behind these special shapes and why there is a limited number of regular polyhedra.  How to construct regular polygons and polyhedra including the using the Zometool.

  The history of the Platonic solids and the geometry of perfection. Where and why we find regular polygons and polyhedra in the natural world.

Stuff You’ll Need…

 Zometool with Green Lines

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ePortfolio Suggestions…

     Before you begin the activities in this Learning Launcher write down anything you already know about regular polygons, especially squares, triangles and pentagons.

List the three most interesting things you learned in the

What You Should Know

… section.

Keep a list of glossary words you learn. Pay particular attention to the

bold italicized

words you find in this Learning Launcher.

Take photos of the platonic solids and other shapes you construct and include them in your presentation. Document your process as you experiment with different shapes in attempts to build complex polyhedra.

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What You Should Know…

Intro to Zometool

Zometool is a fun mathematical modeling kit that allows learners to build an endless number of shapes that can be as simple as a triangle or as complex as the models shown below. Zometool designers put a lot of thought into the math behind this tool and the results are impressive!

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What You Should Know…

Intro to Zometool

Zometool is a series of small white spheres called

nodes

varied colored sticks called shapes.

Blue struts struts

. The nodes have three different shapes holes on them: triangles, pentagons, and rectangles. Each color of strut has a tip of one of those have rectangular tips,

yellow struts

and have triangles,

red

and

green struts

have pentagons. The angles between holes in the nodes as well as the lengths of the different struts help illustrate hundreds of math principles.

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What You Should Know…

Intro to Zometool

Strut length:

For each color of strut, there are three sizes. The smaller two sizes, when put together, equal the largest size. It is also the case that if you divided the length of the smallest strut by the medium strut you would get the same number that you get when dividing the medium strut by the large. This is called the

golden ratio,

and is something you can chose to explore in a later Learning Launcher.

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What You Should Know…

Intro to Zometool

The Nodes:

Zome nodes have holes in them that correspond with the tips on the struts. They are aligned on the node in such a way that shapes made with different colored struts have varied types of symmetry. For example, two-dimensional objects made perpendicular to blue struts tend to have two-fold symmetry.

Use the holes as a guide for building your models. If you aren’t sure which color strut to use in your model, look down the line between the nodes and “sight” the hole. Here, it looks like we need a blue strut with a rectangular tip!

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What You Should Know…

Polygons

The Platonic Solids are made from

regular polygons

. Before we get to talking about the solids, we should go over what a regular polygon is.

Polygons

are made with straight lines that complete a loop, or create closed shape.

Can you tell which of the objects below are not polygons?

This shape is not a polygon since it is open and the lines don’t all connect.

This “teardrop is not a polygon since it has a curved section. All lines in a polygon need to be straight.

The rest of these objects are polygons though one is concave . We’ll talk about this in the next slide.

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What You Should Know…

Convex vs. Concave Polygons

The Platonic Solids are made from

convex polygons

.

Convex polygons

have all internal angles less than 180 o . In other words, they don’t have any “bites” taken out of them.

Concave polygons

do have internal angles greater than 180 o and have “bites”. This polygon is

convex

since all internal angles are less than 180 o .

This polygon is

concave

since at least one internal angle is more than 180 o .

Here are a few more concave polygons.

270 o Here are some more convex polygons.

45 o 90 o 45 o

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What You Should Know…

Equilateral Polygons

Equilateral polygons

have all sides of equal length. Can you tell which of these polygons are equilateral?

These polygons are not equilateral – some of the sides are longer than others. We’ll talk about this on the next slide. equilateral – © Creative Learning Systems www.creativelearningsystems.com

What You Should Know…

Equiangular Polygons

Polygons that have all equal angles are called

equiangular

. Can you tell which of the shapes below are equiangular?

they are not o

90 o 60 o 108 o

These polygons equiangular. All of their angles have the same value.

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What You Should Know…

Regular Polygons

Regular polygons are both equilateral AND equiangular. Since concave polygons can’t be equiangular, all regular polygons are also convex.

Equiangular Polygon

All angles are the same.

90 o

Regular Polygon

90 o

Equilateral Polygon

All sides are the same length.

This is a regular polygon because it is both equilateral and equiangular.

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What You Should Know…

Regular Polygons

Still a bit confused? Try watching this video explaining regular polygons.

Tutorial Video: Regular Polygons © Creative Learning Systems www.creativelearningsystems.com

What You Should Know…

Regular Polygons

As long as the sides and angles are equal the polygon is regular. Here are some of the smaller ones.

Regular or Equilateral Triangle Square or Regular Quadrilateral Regular Pentagon Regular Hexagon Regular Heptagon Regular Octagon 3

equal sides All angles 60 o

4

equal sides All angles 90 o

5

equal sides All angles 108 o

6

equal sides All angles 120 o

7

equal sides All angles 128.5

o

8

equal sides All angles 135 o

Prove It!

We often refer to a regular triangle as an equilateral triangle. They are the same thing. If you think about it, an equilateral triangle has to be equiangular as well. But don’t take my word for it! Try to make a triangle with Zome struts that has three struts of the same length but different angles.

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What You Should Know…

Regular Polygons

…and you can keep going! There is an infinite number of regular polygons! As you add more and more sides, the shape looks more and more like a circle and the angle gets closer and closer to 180 o .

Regular Decagon Regular Icosagon 10

equal sides All angles

144 o 20

equal sides All angles

162 o © Creative Learning Systems www.creativelearningsystems.com

What You Should Know…

The Geometry of Perfection

Scholars before Ancient Greek mathematicians knew of regular polygons. However, it was the Greeks like Pythagoras and Plato that started to ponder the perfection of three dimensional geometry. (Three dimensional shapes that use polygons and are closed are called

polyhedra

.) It is clear that an equilateral triangle or square is “perfect”, but what about perfect polyhedra? It is the pursuit of three dimensional perfection that led these mathematicians to the study of the

Platonic Solids

. We will introduce the platonic solids on the next slide, but first, think about what a “perfect” three dimensional polyhedron would look like. What types of polygons would you use to construct them? What “rules” would you have?

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What You Should Know…

The Platonic Solids – What makes a Platonic Solid?

Hopefully you though about what might make a “perfect” polyhedron. The Greeks put a lot of time into this idea as well. Here are the characteristics of a perfect polyhedron, or Platonic Solid:

1. They are Made from one type of regular polygon.

2.All polygons are the same size.

3.The same number of polygons come together at each point (vertex).

These seem like pretty sensible rules to start with. The question is, how many objects can you make that meet these qualifications? Is it infinite like the number of regular polygons or is the number limited? The next few slides will investigate these questions.

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What You Should Know…

The Platonic Solids – How many are there?

Let us start our proof by thinking about what goes into making a polyhedron in general. Some terms used in describing polyhedra are: •

Face(s)

The two-dimensional polygons that make up a polyhedron •

Edges

The sides of the polygons in a polyhedron •

Vertex or vertices

These are the points on polyhedron where the edges come together.

Notice how two faces come together on each edge and each vertex has three edges coming to a point . You can have more than three coming together at a vertex, but no less. (Two edges coming meeting at a point ether only makes a two dimensional object.) © Creative Learning Systems www.creativelearningsystems.com

What You Should Know…

The Platonic Solids – How many are there?

Let’s start by trying to figure out how many Platonic solids can be made with squares as the faces. Four squares coming together at a point makes a flat surface, like the four tiles on a floor shown below. Another way of thinking of it is, each corner of a square is 90 o . 4x90 o =360 o and 360 o is all the way around a point. You need the faces to add up to less than 360 360 o o for them to bend in different dimensions and make a polyhedron. Any sum of angles more than means there isn’t room for all the polygons.

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What You Should Know…

The Platonic Solids – How many are there?

So, if you need at least three faces to come together to make a polyhedron and four squares makes a flat surface then

together at a point. the only polyhedron you can make with squares has three coming

In the same room with the tile floor, we see three squares made by two walls and the floor coming together at a vertex in the corner making the polyhedron that is the room. This is our first platonic solid, struts!

a cube

! Try building a cube using blue

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What You Should Know…

The Platonic Solids – How many are there?

A cube

is the only Platonic solid that can be made using squares. It has three squares coming together at each vertex. Any less than three squares wouldn’t make a vertex, four squares makes a flat surface and five or more won’t fit. Notice that a cube has six faces and eight vertices. Keep track of the number of faces and vertices as we continue to discover the platonic solids… It will come in handy when we talk about duals later.

any length You can build a cube with blue struts . Give it a shot!

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What You Should Know…

The Platonic Solids – How many can be made with triangles?

Let’s look at the platonic solids we can make using

equilateral triangles

. Remember that each angle on a equilateral triangle is 60 make a polyhedron and 3 x 60 o =180 o o . We know we need at least three triangles to which is less than 360 o so we know three equilateral triangles will fit around a vertex, (with plenty of space to spare.) What shape do you get when you take three equilateral triangles and then fold them down such that the edges meet?

and building a polyhedron with three equilateral triangles coming together at each vertex.

end up with a polyhedron with 4 faces and 4 vertices.

Try using green struts

When you finish you should

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What You Should Know…

The Platonic Solids – How many can be made with triangles?

We can also make a platonic solid with four equilateral triangles since 4 x 60 o =240 o which is still less than 360 o . This shape is a little more complicated than the one you make with three, but all you need to remember is every vertex in a platonic solid looks exactly the same.

Try using the green struts and making four equilateral triangles that come together at a vertex.

You should have a pyramid with a square base. Unfortunately, this is not a platonic solid since all polygons need to be the same and we are working with triangles, not squares. The solution to your problem is to

bring four triangles together at every vertex

. This famous sculpture by I.M. Pei at the Louver in Paris, France is the platonic solid with four equilateral triangles coming together at each vertex. It looks like a pyramid, but there is something below ground too… When you finish you should have a polyhedron with 8 faces and 6 vertices.

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What You Should Know…

The Platonic Solids – How many can be made with triangles?

We can also make a Platonic solid with five equilateral triangles since 5 x 60 o =300 o which is still a little bit less than 360 o .

Try using the blue struts and building a shape with five equilateral triangles coming off the same point. Repeat this for every vertex you make and see what you get! There are five equilateral triangles coming together at this vertex.

you are done, your polyhedron should have 20 faces and 12 vertices! © Creative Learning Systems www.creativelearningsystems.com

What You Should Know…

The Platonic Solids – How many can be made with triangles?

What about six equilateral triangles? 6 x 60 o =360 vertex. o which means our surface is flat, just like we had with four squares put together. This tells us that you can’t make a Platonic solid using 6 or more equilateral triangles. Six equilateral triangles at one vertex is a flat object and seven or more won’t fit around a So, now we have four platonic solids.

•The cube (3 squares at each vertex) •3 equilateral triangles at each vertex •4 equilateral triangles at each vertex •5 equilateral triangles at each vertex Is that it? Well, let’s keep our investigation going a little longer…

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What You Should Know…

The Platonic Solids – Pentagons?

We have figured out all the platonic solids possible with regular three-sided polygons (triangles), and regular four sided polygons (squares). The next step is looking at a five sided regular polygon,

the regular pentagon

.

Each angle in a regular pentagon is 108 o . Since 3 x 108 o = 324 o and 324 o is less than 360 with 12 faces and 20 vertices. o , we know we can make a platonic solid with three regular pentagons coming together at each vertex

. Try using the blue struts and building a polyhedron using three pentagons at each vertex.

When you finish, you should end up with a large polyhedron Since we could just barely get three pentagons together, there is no way we could make a polyhedron using four pentagons at each vertex, (4 x 108 o = 432 o > 360 o ) so there are no other Platonic solids possible with pentagons.

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What You Should Know…

The Platonic Solids – Hexagons?

So we have three Platonic solids with triangles, one with squares, and one with pentagons sides?

– what about hexagons or regular polygons with more Each angle inside a hexagon is 120 hexagons coming together at a vertex make a flat surface (3 x 120 o = 360 o ). This means we cannot make any Platonic solids with hexagons. All regular polygons with more sides than hexagons, like heptagons have obtuse enough angles that three won’t fit around a vertex. o so three This tells us we can’t make platonic solids with hexagons or larger regular polygons.

Three hexagons together make a flat surface. This beehive and this table are made of hexagons.

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What You Should Know…

The

Five

, and Only Five, Platonic Solids.

It turns out that there are only five Platonic solids . The process we just went through is what you call a and combinations of the three.

geometric proof

. As long as our logical arguments are correct, we have proven that there are no more possible Platonic solids. Professional mathematicians work to prove things all the time and have been doing this since ancient times. Some proof use geometry, others algebra, others calculus So here they are: Tetrahedron Octaherdon Icosahedron Cube Dodecahedron (3 Triangles) (4 Triangles) (5 Triangles) (3 Squares) (3 Regular Pentagons)

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What You Should Know…

The History of the Platonic Solids

Mathematicians have known about the Platonic Solids for thousands of years. There is some evidence that Pythagerous identified at least some of them around 500 B.C, and Theaeteus contributed the last few around 150 years later and also proved that there were no more than five.

Ancient Platonic Solid stones found in Scotland

Euclid wrote

Elements

around 300 B.C. and explained many geometric principles still taught in math courses today. It is estimated that

Elements

is second only to the Bible in editions published. He devotes an entire book to explanation and proof of the Platonic solids.

Euclid’s Elements

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What You Should Know…

The Geometry of Perfection

The Platonic Solids are sort of “ideal” shapes. Everything is the same. The ancient Greeks were very interested in the idea of mathematical perfection

. Plato

, established the

Theory of Forms

describing perfect forms of objects as possessing the true essence of items. For example, there is an ideal “form” of an apple that all apples are somewhat like, but none are exactly like it. Every circle drawn is imperfect, but they all are attempts at the “form” of a circle. Plato, Euclid, Pythagoras and other Greeks were very interested in the mathematics of perfection – the math behind perfect forms. It isn’t surprising that with this focus on ideal forms that Plato studied the Platonic Solids.

Greek Mathematician, Pythagoras Greek Philosopher, Plato

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What You Should Know…

The Platonic Solids and the Elements

Plato, whom these shapes are named after, tried to explain the universe in terms of these perfect shapes. He associated the four elements, earth, wind, fire and water with these regular convex polyhedra. Plato associated the fifth Platonic Solid, the dodecahedron, with the shape of the universe itself. Earth Wind Fire Water The Universe

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What You Should Know…

Kepler and the Motion of the Planets

In the late 1500s and early 1600s astronomers were engaged in a lively debate about what was at the center of our solar system. Nicolaus Copernicus, in 1543, had proposed a controversial model of the solar system with the sun, instead of the earth, at the center. Much debate followed with most discoveries supporting the findings of Copernicus. German astronomer, Johannes Kepler contributed to the debate by attempting to explain the motion of the planets. Kepler developed an elegant hypothesis that God had placed the planets on the Platonic Solids set inside one another. Years later, his own data caused him to abandon this idea since he found the orbits of the planets were elliptical and not circular.

Kepler’s early model of the solar system involving the Platonic Solids Thanks in part to Kepler’s calculations, we now know that planets orbit in ovals, or ellipses, not circles.

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What You Should Know…

Chance and the Platonic Solids

For many years, people have used the platonic solids in games of chance. Since all of the faces of the platonic solids are the same, they make perfect dice with each face having equal chance of landing up. Obviously, the most common Platonic Solid as a die is the 6-sided cube, but others show up in modern role playing games and have been used throughout history.

What’s the deal? There are only five platonic solids yet there are seven golden dice shown here? Can you figure out which ones aren’t Platonic Solids? Are they still “fair”?

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Do It!

Build the Platonic Solids with Zometool

Now it’s your turn…

Use the Zometool and build each of the Platonic Solids. Here are a few hints: •

Cube

– Use blue struts of equal length. •

Tetrahedron –

Use three green struts at each vertex. It is a tight fit so be careful!

•

Octahedron

– Use four green struts at each vertex. •

Icosahedron –

Use blue struts and five at each vertex. It might be easiest to build a pentagon to start then connect five triangles above it and continue from there.

•

Dodecahedron –

Use blue struts and make sure you aren’t building hexagons!

Tutorial Video: Building the Platonic Solids with Zometool This video will explain why there are only five platonic solids and help guide you as you build them with the Zometool.

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Do It!

Exploring the Relationships Between the Platonic Solids-- Duals

Now it’s your turn…

Take a look at your Platonic Solids and fill out the table below. (A few are done for you to get you started.) Do you notice any similarities between them? Are there some that seem like “opposites” of each other? Once you fill out the table, watch the tutorial and build the platonic duals!

Platonic Solid

Tetrahedron

Faces

4

Vertices Watch this tutorial video, then try to build the solids with their duals.

Octahedron 6 Cube Icosahedron Dodecahedron

Tutorial Video: Building the Platonic Solids and their Duals © Creative Learning Systems www.creativelearningsystems.com

Extend Yourself…

The Archimedean Solids…

Remember the rules for building Platonic Solids? You need to use the same polygon throughout. There are a second group of regular polyhedra that use two or more different regular polygons. These are called the

Archimedean solids

and there are thirteen of them. One you might be familiar with is a traditional soccer ball, or truncated icosahedron. Eleven of these are possible with Zometool using blue and green struts. How many can you build?



Extend Yourself…

Plato associated the Pplatonic Solids with the four elements (air, fire, earth, water) and the structure of the universe. Make some “ZomeArt” by using the

Vanishing Point

tool in

Adobe Photoshop

to past correctly oriented pictures of the different elements over photos of your Platonic solids.

Here’s an example of an icosahedron with water images. This was developed using the Vanishing Point tool in Adobe Photoshop.

  

Extend Yourself…

Are you up for a historic challenge? It is possible to build all the Platonic Solids in one shape with each inside another – Much like Kepler tried to do with the planets. It isn’t easy though! Try starting with the icosahedron in the middle… In the Zometool green line instructions you have in your SmartLab, it explains how to build five tetrahedra inside a dodecahedron. There are also other models explained that explore the relationships between platonic solids such as a truncated tetrahedron and octahedron. Try building as many as you can.

The Platonic solids are found in the natural world and cubes are often found in crystal structures. physical world. Include some examples in your presentation.

Here’s a model

– some more than others. Octahedrons, tetrahedrons Icosahedrons are the shape of many viruses! Spend some time researching the solids in the natural and

of an icosahedral virus. Why do you think they are in this shape?

Kepler’s model of the solar system