Investigating Regular Polyhedra Level 1 What Youll 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 ID: 807035
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Slide1
Three-dimensional Geometry
Investigating Regular Polyhedra– Level 1
Slide2What 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
Slide3ePortfolio 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.
Slide4What 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!
Slide5What You Should Know…
Intro to Zometool
Zometool is a series of small white spheres called
nodes
and varied colored sticks called 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 shapes.
Blue struts have rectangular tips, yellow struts
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.
Slide6What 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.
Slide7What 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!
Slide8What 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 “teardrop is not a polygon since it has a curved section. All lines in a polygon need to be straight.
This shape is not a polygon since it is open and the lines don’t all connect.
The rest of these objects are polygons though one is
concave
. We’ll talk about this in the next slide.
Slide9What You Should Know…
Convex vs. Concave Polygons
The Platonic Solids are made from
convex polygons
. Convex polygons have all internal angles less than 180o. In other words, they don’t have any “bites” taken out of them. Concave polygons do have internal angles greater than 180o
and have “bites”.
270
o
90
o
45
o
45
o
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 some more convex polygons.
Here are a few more concave polygons.
Slide10What 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
equilateral
– all sides are equal length.
These polygons are not equilateral– some of the sides are longer than others.
Even though it isn’t equilateral, there is something special about this rectangle. It is
equiangular
. We’ll talk about this on the next slide.
We use these hash marks to indicate sides are the same length.
Slide11What You Should Know…
Equiangular Polygons
Polygons that have all equal angles are called
equiangular
. Can you tell which of the shapes below are equiangular?
These polygons equiangular. All of their angles have the same value.
90
o
108
o
60
o
These polygons have different internal angles so they are not equiangular.
If you think about it, concave polygons are never equiangular since some internal angles are > 180
o
and not all internal angles can be > 180
o
or the polygon would never close!
Slide12What 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.
90
o
Equiangular Polygon
All angles are the same.
Equilateral Polygon
All sides are the same length.
90
o
Regular Polygon
This is a regular polygon because it is
both
equilateral and equiangular.
Slide13What You Should Know…
Regular Polygons
Still a bit confused? Try watching this video explaining regular polygons.
Tutorial Video: Regular Polygons
108
o
60
o
Slide14What 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.
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
Regular or Equilateral
Triangle
Square or Regular Quadrilateral
Regular Pentagon
Regular Hexagon
Regular Heptagon
Regular Octagon
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.
Slide15What 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
.
10
equal sides
All angles 144o
20 equal sidesAll angles
162
o
Regular Decagon
Regular Icosagon
Slide16What 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?
Slide17What You Should Know…
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:
They are Made from one type of regular polygon.
All polygons are the same size.
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.
The Platonic Solids– What makes a Platonic Solid?
Slide18Notice 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.)
What You Should Know…
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 polyhedronEdges The sides of the polygons in a polyhedron
Vertex or vertices These are the points on polyhedron where the edges come together.
The Platonic Solids– How many are there?
Slide19What 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
o
for them to bend in different dimensions and make a polyhedron. Any sum of angles more than 360
o
means there isn’t room for all the polygons.
Slide20What 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
the only polyhedron you can make with squares has three coming together at a point.
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,
a cube
! Try building a cube using blue struts!
Slide21What 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.
You can build a cube with any length
blue struts
. Give it a shot!
Slide22What 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
o. We know we need at least three triangles to make a polyhedron and 3 x 60
o=180o 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? Try using
green struts and building a polyhedron with three equilateral triangles coming together at each vertex.
When you finish you should end up with a polyhedron with 4 faces and 4 vertices.
60
o
60
o
60
o
Slide23What 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.
60
o
60
o
60
o
60
o
Slide24What 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!
60
o
60
o
60
o
60
o
60
o
There are five equilateral triangles coming together at this vertex.
Keep building the polyhedron by bringing five equilateral triangles together at all the other vertices as well,
Keep building the polyhedron by bringing five equilateral triangles together at all the other vertices as well. When you are done, your polyhedron should have 20 faces and 12 vertices!
Slide25What You Should Know…
The Platonic Solids– How many can be made with triangles?
What about six equilateral triangles? 6 x 60
o
=360
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 vertex.
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…
Slide26What 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 108o = 324o and 324
o is less than 360
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 with 12 faces and 20 vertices.
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.
108
o
108
o
108
o
Slide27What You Should Know…
The Platonic Solids– Hexagons?
So we have three Platonic solids with triangles, one with squares, and one with pentagons– what about hexagons or regular polygons with more sides?
Each angle inside a hexagon is 120
o
so three hexagons coming together at a vertex make a flat surface (3 x 120
o
= 360o). 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.
Three hexagons together make a flat surface. This beehive and this table are made of hexagons.
This tells us we can’t make platonic solids with hexagons or larger regular polygons.
Slide28What 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 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 and combinations of the three.
So here they are:
Tetrahedron Octaherdon Icosahedron Cube Dodecahedron
(3 Triangles) (4 Triangles) (5 Triangles) (3 Squares) (3 Regular Pentagons)
Slide29What 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.
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.
Ancient Platonic Solid stones found in Scotland
Euclid’s Elements
Slide30What 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
Slide31What 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
Slide32What 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.
Slide33What 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”?
Slide34Do 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.
Slide35Do 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 SolidFaces
Vertices
Tetrahedron4
Octahedron
6
Cube
Icosahedron
Dodecahedron
Watch this tutorial video, then try to build the solids with their duals.
Tutorial Video: Building the Platonic Solids and their Duals
Slide36Extend 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?
Slide37Extend 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.
Slide38Extend 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– some more than others. Octahedrons, tetrahedrons and cubes are often found in crystal structures. Icosahedrons are the shape of many viruses! Spend some time researching the solids in the natural and physical world. Include some examples in your presentation.
Here’s a model of an icosahedral virus. Why do you think they are in this shape?
Kepler’s model of the solar system