PlatonicPlatonic

SolidsSolids

So What is Our Topic?So What is Our Topic?

It’s math (go figure!)It’s math (go figure!) It’s hands on (so you get to play!) It’s hands on (so you get to play!) There’s candy involvedThere’s candy involved There are some objects we will describeThere are some objects we will describe

that are named after a famous personthat are named after a famous person

because everything has to be named something, because everything has to be named something, right??right??

You know how this goes—here’s some real life You know how this goes—here’s some real life examples of what I meanexamples of what I mean

Air Jordans

                          

         

Tigerade

                              

                 

Platonic Solids

What are these Platonic Solids?What are these Platonic Solids?

Regular three dimensional solids called Regular three dimensional solids called polyhedra (because they are made from polyhedra (because they are made from polygons)polygons)

All the faces (sides) are regular polygonsAll the faces (sides) are regular polygons All the vertices (corners) look exactly the All the vertices (corners) look exactly the

samesame ConvexConvex There are only fiveThere are only five

http://math.ucr.edu/home/baez/platonic.htmlhttp://math.ucr.edu/home/baez/platonic.html

http://illuminations.nctm.org/http://illuminations.nctm.org/ActivityDetail.aspx?id=70ActivityDetail.aspx?id=70

Some definitions—so we all Some definitions—so we all know what we’re talking about!know what we’re talking about!

PolygonPolygon- any figure that is closed (connected to itself) and has three - any figure that is closed (connected to itself) and has three or more sides that are straight, not curvedor more sides that are straight, not curved

Plural is polygonsPlural is polygons These are flat (two dimensional)These are flat (two dimensional)

Examples of polygons

Examples of figures that are not polygons

Some definitions—so we all Some definitions—so we all know what we’re talking about!know what we’re talking about!

Regular Polygon - a polygon has the same side lengths and same angle measures

Examples of regular polygons

Examples of figures that are not regular polygons

Some definitions—so we all Some definitions—so we all know what we’re talking about!know what we’re talking about!

Polyhedron-Polyhedron- a solid figure made up of polygons a solid figure made up of polygons Plural is Plural is polyhedrapolyhedra These are not flat (three dimensional)These are not flat (three dimensional)

Would the polyhedra above ALL be considered regular?

Some definitions—so we all Some definitions—so we all know what we’re talking about!know what we’re talking about!

Convex-Convex- a shape where the surfaces bulge outwards a shape where the surfaces bulge outwards The opposite of this is The opposite of this is concaveconcave

Examples of concave figuresExamples of convex figures

Some definitions—so we all Some definitions—so we all know what we’re talking about!know what we’re talking about!

Vertex-Vertex- where three or more faces meet where three or more faces meet Plural is Plural is verticesvertices Hint—it’s the cornersHint—it’s the corners

Face-Face- the side the sideEdgeEdge- where two faces (sides) meet- where two faces (sides) meet

Forms a line segmentForms a line segment

Euler’s Formula--Cool Rule!Euler’s Formula--Cool Rule!

Faces + Vertices - Edges = 2 Faces + Vertices - Edges = 2

This is true for all convex This is true for all convex polyhedrapolyhedra

Not just true for platonic Not just true for platonic solidssolids

1752

What is a Net?What is a Net?

If you had a 3D shape made out of paper, If you had a 3D shape made out of paper, that was all folded and taped together, and that was all folded and taped together, and you pulled the tape off and laid the figure you pulled the tape off and laid the figure out flat, then you would get the net of the out flat, then you would get the net of the three dimensional solid. three dimensional solid.

There’s more than one net for each solid.There’s more than one net for each solid.

http://www.ask.com/bar?http://www.ask.com/bar?q=what+is+a+net+in+geometry&page=1&q=what+is+a+net+in+geometry&page=1&qsrc=19&ab=2&u=http%3A%2Fqsrc=19&ab=2&u=http%3A%2F%2Fwww.mathsnet.net%2Fgeometry%2Fwww.mathsnet.net%2Fgeometry%2Fsolid%2Fnets.html%2Fsolid%2Fnets.html

Your turnYour turn

Using the plastic squares and triangles Using the plastic squares and triangles Make as many nets as possible for a cube Make as many nets as possible for a cube

and tetrahedronand tetrahedron Draw ALL of your nets on the grid paperDraw ALL of your nets on the grid paper Hint---there are eleven for the cube and Hint---there are eleven for the cube and

two for the tetrahedron!!two for the tetrahedron!! First one to get all eleven for the cube First one to get all eleven for the cube

gets a candy bargets a candy bar Duplicates don’t count!Duplicates don’t count!

No duplicates!No duplicates!

Let’s look at the netsLet’s look at the nets

http://illuminations.nctm.org/http://illuminations.nctm.org/activitydetail.aspx?ID=84activitydetail.aspx?ID=84

http://www.numeracycd.com/contents/http://www.numeracycd.com/contents/activities/nets/netscube+puzzle.pdfactivities/nets/netscube+puzzle.pdf