Sketchup Mathematical Activities

Exploring 2D Design

I. Polygons and 2D Symmetry

1. Rotational Symmetry

2. Mirror Symmetry

II. Frieze Patterns

III. Tesselations

1. Show all triangles tessellate

2. Show all quadrilaterals tessellate

3. The pentagon story

4. Investigate Regular, Semi-regular and Demi-Regular Tesselations

5. Create Escher like Tesselations

6. The 17 Wallpaper Designs

IV. Aperiodic Tilings

http://web.media.mit.edu/~black/tiles/aperiodic.html

1. Some special tiles

a. Hirschhorn

b. Voderberg

http://www.uwgb.edu/dutchs/symmetry/radspir1.htm

c. Spidron

2. Penrose Tiles

a. Rhombs

http://www.uwgb.edu/DutchS/SYMMETRY/penrose.htm

b. Kites and Darts

3. Islamic Girih Tiles

4. The Mitre System

Exploring 3D Design

1. Prisms and Pyramids

a. Prism Nets

b. Pyramid Nets

2. 3D Symmetry

3D Symmetry is defined in terms of rigid motions that leave a figure unchanged. These motions are always rotations about an axis. Using SketchUps Rotation Tool one can show all the different axes of symmetry for a 3D object

3. Constructing Regular Polyhedra ( Platonic Solids )

1. Cube, Tetrahedron and Octahedron

a. Tetra in Cube

b. Octahedron in a Tetrahedron in A Cube

c. Tetrahedron made up of 4 Smaller Tetras and One Octahedron

d. Building Tetrahedra, Octahedra and Cubes from Nets

e. Exploring Duality

2. Icosahedron and Dodecahedron

a) Every vertex of an Icosahedron lies on one of three Golden Rectangles as shown below

b) Every Vertex of a Dodecahedron lies on either the vertices of a cube with sides x or on one of three rectangles with sides of length gx and x/q where g is the golden ratio.

d. Building Icosahedra and Dodecahedra from Nets

e. Exploring Duality

4. Constructing Semi-Regular Polyhedra

( Archimedean Solids )

a) Using Truncation to Construct The Truncated Cube, The Truncated Octahedron and The Cuboctahedron

b) Using Symmetry and Some Trig to Construct the Rhombicuboctahedron and the Rhombi Truncated Cuboctahedron

c) Using Truncation to Construct The Truncated Icosahedron, The Truncated Dodecahedron and The Icosadocecahedron

e) Using Symmetry and Some Trig to Construct the Rhombicosadodecahedron and the Rhombi Truncated Icosadocecahedron.

f) Using Symmetry and Some Trig to Construct the Snub Cube and the Snub Dodecahedron.

5. Constructing Archimedean Duals

( Catalan Solids )

6. Constructing Compounds of Cubes and Tetrahedra

7. Tetrahedral Geometry

a) Constructing the Incenter, the Circumcenter, the Centroid and The Orthocenter when it exists.

b) Exploring The Amazing Properties of Orthocentric Tetrahedra

A tetrahedron is orthocentric if the four altitudes are convergent (existence of an orthocentre H) or the three pairs of opposite edges are orthogonal (characteristic property)

• the feet of the altitudes are orthocentres of the faces

• the three common perpendicular to opposite edges are convergent in H

• the three segments joining the midpoints of opposite edges have same length

• the midpoints of the edges and the feet of the common perpendiculars to opposite edges lay on a sphere with centre the isobarycentre G of the vertices (first Euler's sphere)

• with O centre of the circumscribed sphere, G is midpoint of [OH] (Euler's line of the tetrahedron)

• the perpendiculars to the faces in their centres of gravity are convergent in I on the Euler's line

• in a tetrahedron ABCD, the feet of the altitudes, the centres of gravity of the faces, and the points laying on the thirds of [HA], [HB], [HC] and [HD] lay on a sphere with centre the midpoint of [HI] (second Euler's sphere)

• in a tetrahedron ABCD, AB²+ CD² = AD²+ BC²

examples: the regular tetrahedra, the trirectangle tetrahedral