Download 2D and 3D Design Notes

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
http://britton.disted.camosun.bc.ca/jbperplex.htm
http://www.mathpuzzle.com/tilepent.html
4. Investigate Regular, Semi-regular and Demi-Regular
Tesselations
http://mathworld.wolfram.com/Tessellation.html
5. Create Escher like Tesselations
http://www.kmlhs.org/faculty/TKUEHL/Precalculus/MESA/
Tessellations/TK%20Tessellation%20Home.htm
6. The 17 Wallpaper Designs
IV. Aperiodic Tilings
http://web.media.mit.edu/~black/tiles/aperiodic.html
1. Some special tiles
a. Hirschhorn
http://burtleburtle.net/bob/other/bathroom.html
b. Voderberg
http://www.uwgb.edu/dutchs/symmetry/radspir1.htm
c. Spidron
http://www.szinhaz.hu/spidron/
2. Penrose Tiles
a. Rhombs
http://www.uwgb.edu/DutchS/SYMMETRY/penrose.htm
b. Kites and Darts
http://mathworld.wolfram.com/PenroseTiles.html
3. Islamic Girih Tiles
http://en.wikipedia.org/wiki/Girih_tiles
4. The Mitre System
http://www.mathpuzzle.com/Mitre2.html
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
http://www.ac-noumea.nc/maths/amc/polyhedr/tetra_.htm