Polyhedra Helmer ASLAKSEN Department of Mathematics

Slides:

Advertisements

Similar presentations

Tilings and Polyhedra Helmer ASLAKSEN Department of Mathematics National University of Singapore

Advertisements

Geometry Mini-Lesson Below is the net of a polyhedron. How many vertices does the polyhedron have? MA.912.G.7.1: Describe and make regular, non-regular,

Convex Polyhedra with Regular Polygonal Faces David McKillop Making Math Matter Inc.

Faces, Vertices and Edges

EXAMPLE 1 Identify and name polyhedra

4.5 More Platonic Solids Wednesday, March 3, 2004.

4.5 Platonic Solids Wednesday, February 25, 2009.

Chapter 12 Surface Area and Volume. Topics We Will Discuss 3-D Shapes (Solids) Surface Area of solids Volume of Solids.

Geometry Polyhedra. 2 August 16, 2015 Goals Know terminology about solids. Identify solids by type. Use Euler’s Theorem to solve problems.

The Beauty of Polyhedra Helmer ASLAKSEN Department of Mathematics National University of Singapore

Chapter 12 Surface Area and Volume. Topics We Will Discuss 3-D Shapes (Solids) Surface Area of solids Volume of Solids.

Surface Area and Volume

 A Polyhedron- (polyhedra or polyhedrons)  Is formed by 4 or more polygons (faces) that intersect only at the edges.  Encloses a region in space. 

Chapters 5, 6, 9 : Measurement and Geometry I and II

Surface Area and Volume Chapter 12. Exploring Solids 12.1 California State Standards 8, 9: Solve problems involving the surface area and lateral area.

GEOMETRY Bridge Tips: Be sure to support your sides when you glue them together. Today: Over Problem Solving 12.1 Instruction Practice.

SOLID FIGURES SPI

Geometry: Part 2 3-D figures.

Name the polygon by the number of sides.

Polyhedrons or Polyhedra A polyhedron is a solid formed by flat surfaces. We are going to look at regular convex polyhedrons: “regular” refers to the fact.

Right Prisms & Cylinders, Right Pyramids & Cones, Platonic Solids, Composite Figures.

How many vertices, edges, and faces are contained in each of the polyhedra? vertices of each polygon polygons meeting at a vertex faces of the polyhedron.

6-3B Regular Polyhedrons

3-Dimentional Figures Section 11.1.

A regular polygon is a polygon with all sides congruent and all angles congruent such as equilateral triangle, square, regular pentagon, regular hexagon,

1)The locus of points, lying in a plane, that are equidistant from a specific point – the center. 2)A regular polygon with an infinite number of sides.

Platonic Solids. Greek concept of Symmetry Seen in their art, architecture and mathematics Seen in their art, architecture and mathematics Greek Geometry.

Chapter 12 Section 1 Exploring Solids Using Properties of Polyhedra Using Euler’s Theorem Richard Resseguie GOAL 1GOAL 2.

12.1– Explore Solids.

Frameworks math manipulative by Rob Lovell. Frameworks math manipulative Rob Lovell Contents What are Frameworks? How can a teacher use them? Why would.

Beauty, Form and Function: An Exploration of Symmetry

Do you know what these special solid shapes are called?

Three-Dimensional Figures – Identify and use three-dimensional figures.

12.1 – Explore Solids.

6-3A Geometry Section 6-3B Regular Polyhedrons Page 448 If you have your solids, you might like to use them today. Test Friday – Shapes on Friday On.

Platonic Solids MATH 420 Presentation: Kelly Burgess.

Chapter 12.1 Notes Polyhedron – is a solid that is bounded by polygons, called faces, that enclose a single region of space. Edge – of a polygon is a line.

12.1 & 12.2 – Explore Solids & Surface Area of Prisms and Cones.

Year 10 Advanced Mathematics or Convex Regular Solids PLATONIC SOLIDS More correctly known as 

Ch 12 and 13 Definitions. 1. polyhedron A solid with all flat surfaces that enclose a single region of space.

1 Faces, Edges and Vertices Press Ctrl-A ©2009 G Dear – Not to be sold/Free to use Stage 4 Years 7 & 8.

Section 12-1 Exploring Solids. Polyhedron Three dimensional closed figure formed by joining three or more polygons at their side. Plural: polyhedra.

Space Figures and Nets Section 6-1 Notes and vocabulary available on my home page.

Types of 3D Shapes Slideshow 42, Mathematics Mr Richard Sasaki Room 307.

Diamond D’Oveyana & Sylvia

3-D Geometry By: _____. Platonic Solids These platonic solids were made with Zometools. A platonic solid is _____ There are five platonic solids.

Polyhedron Models By students of Dean Zeller Pre-Algebra, Algebra, Geometry Kansas City Missouri School District CS1101 Art Institute of Jacksonville.

Surface area and volume. Polyhedrons Polyhedron- a 3-dimensional object with all flat surfaces.

12.1 Exploring Solids Geometry. Defns. for 3-dimensional figures Polyhedron – a solid bounded by polygons that enclose a single region of shape. (no curved.

G.3.J Vocabulary of Three-Dimensional Figures

Platonic Solids And Zome System.

Geometric Solids POLYHEDRONS NON-POLYHEDRONS.

Goal 1: Using Properties of Polyhedra Goal 2: Using Euler’s Theorem

Polyhedra and Prisms.

Polyhedra Mikhilichenko Yelena-Maths teacher

Polyhedra Mikhаilichenko Yelena-Maths teacher

Chapter 11 Extending Geometry

Measurement of Solids & Figures

Nets and Categorising 3D Shapes

12-1 Properties of Polyhedra

Mehmet Kemal ER Seray ARSLAN

Surface Area and Volume

Vertical Angles Vertical angles are across from each other and are created by intersecting lines.

Geometry Chapter : Exploring Solids.

14 Chapter Area, Pythagorean Theorem, and Volume

11.4 Exploring Solids Geometry How many geometric solid can you name?

Presentation transcript:

Polyhedra Helmer ASLAKSEN Department of Mathematics National University of Singapore aslaksen@math.nus.edu.sg www.math.nus.edu.sg/aslaksen/polyhedra/

What is a polygon? Sides and corners. Regular polygon: Equal sides and equal angles. For n greater than 3, we need both.

A quick course in Greek 3 4 5 6 7 Tri Tetra Penta Hexa Hepta 8 9 10 12 20 Octa Ennea Deca Dodeca Icosa

More about polygons The vertex angle in a regular n-gon is 180 (n-2)/n. To see this, divide the polygon into n triangles. 3: 60 4: 90 5: 108 6: 120

Polyhedra What is a polyhedron? Platonic solids Deltahedra Archimedean solids Colouring Platonic solids Stellation

What is a polyhedron? Solid or surface? A surface consisting of polygons.

Polyhedra Vertices, edges and faces.

Platonic solids Euclid: Convex polyhedron with congruent, regular faces.

Properties of Platonic solids Faces Edges Vertices Sides of face Faces at vertex Tet 4 6 3 Cub 12 8 Oct Dod 30 20 5 Ico

Colouring the Platonic solids Octahedron: 2 colours Cube and icosahedron: 3 Tetrahedron and dodecahedron: 4

Euclid was wrong! Platonic solids: Convex polyhedra with congruent, regular faces and the same number of faces at each vertex. Freudenthal and Van der Waerden, 1947.

Deltahedra Polyhedra with congruent, regular, triangular faces. Cube and dodecahedron only with squares and regular pentagons.

Archimedean solids Regular faces of more than one type and congruent vertices.

Truncation Cuboctahedron and icosidodecahedron. A football is a truncated icosahedron!

The rest Rhombicuboctahedron and great rhombicuboctahedron Rhombicosidodecahedron and great rhombicosidodecahedron Snub cube and snub dodecahedron

Why rhombicuboctahedron?

Why snub? Left snub cube equals right snub octahedron. Left snub dodecahedron equals right snub icosahedron.

Why no snub tetrahedron? It’s the icosahedron!

The rest of the rest Prism and antiprism.

Are there any more? Miller’s solid or Sommerville’s solid. The vertices are congruent, but not equivalent!

Stellations of the dodecahedron The edge stellation of the icosahedron is a face stellation of the dodecahedron!

Nested Platonic Solids

How to make models Paper Zome Polydron/Frameworks Jovo

Web http://www.math.nus.edu.sg/aslaksen/