Google SketchUp and Geometry
0. SketchUp Basics
a) Basic SetUp
-Go to Window/Model Info and set the units to decimal.
-Go to Window/Styles/Edit/Color/By Material.
b) See Notes for Info about the different tools
1. A Slicing Challenge
Build a cube and then use the Pencil Tool and the Erase Tool to
answer one of Scott Kim’s Boggler puzzles from Discover
Magazine
2. An Interesting Design Problem
The Stanford Puzzle Problem
i) Construct a 200 x 100 x 700 rectangle with a circular hole that
has a diameter of 100 units, an isosceles triangular hole with a
base an a height of 100 units and a square hole 100 units on a
side.
ii) Use the push/pull tool to create a box with a height of 100
units
iii) Create a solid object that will fit through each hole and which
will completely block light.
3. Traditional Constructions
a) Using Circles
Be sure to increase the number of sides.
b) Using Parallels and Perpendiculars
SketchUp will tell you while dragging when a segment under
construction is either parallel or perpendicular to an
existing segment.
4. Some Useful Constructions and
Demonstrations
a) Show that a cube can be dissected into three congruent
tri rectangular pyramids.
b) Show that a regular tetrahedron can be dissected into an
octahedron and four tetrahedron all with the same edge
length. This can be used to show an interesting relationship
between the volume of an octahedron and that of a
tetrahedron.
c) Intersect a cone with a plane to show the different conic
sections.
d) Building a pyramid from a net.
e) Diagram for calculating dihedral angles for polyhedra.
5. A Course in Symmetry
a) Symmetry in 2D
http://www.peda.com/tess/
i) Rosette
http://www.geom.uiuc.edu/java/Kali/program.html
ii) On a Strip
http://www.geom.uiuc.edu/java/Kali/program.html
http://convergence.mathdl.org/images/upload_library/4/vol1/architec
ture/Math/seven.html
iii) In the Plane: Wallpaper Groups
http://www.oswego.edu/~baloglou/103/seventeen.html
http://www.scienceu.com/geometry/articles/tiling/wallpaper.html
iv) Escher’s Wonderful Patterns
http://www.mcescher.com/Gallery/gallery-symmetry.htm
http://www.tessellations.org/
http://library.thinkquest.org/16661/escher/tessellations.1.html
b) Symmetry in 3D
i) Rotational and Mirror
http://www.geom.uiuc.edu/~teach95/kt95/KTL.html
b) Tilings and Tesselations
i) A Polygonal Periodic Approach
1) Regular
http://library.thinkquest.org/16661/simple.of.regular.polygons/regular
.1.html
2) Semi-Regular
http://library.thinkquest.org/16661/simple.of.regular.polygons/semire
gular.1.html
3) Demi-Regular
http://library.thinkquest.org/16661/of.regular.polygons/demiregular.1.
html
2) A Polygonal Aperiodic Approach
http://www.spsu.edu/math/tile/aperiodic/index.htm
i) Hirschhorn and Voderberg Tiles
http://www.uwgb.edu/DutchS/symmetry/radspir1.htm
ii) Penrose Tiles
http://www.ams.org/featurecolumn/archive/penrose.html
iii) The Spidron System
http://www.szinhaz.hu/edan/SpidroNew/
c) Polyhedra
1) Prisms and Pyramids
http://mathworld.wolfram.com/Prism.html
http://mathworld.wolfram.com/Antiprism.html
http://mathworld.wolfram.com/Pyramid.html
2) The Platonic Solids
http://mathworld.wolfram.com/PlatonicSolid.html
http://www.geometrycode.com/sg/polyhedra.shtml
3) The Archimedean Solids
http://mathworld.wolfram.com/ArchimedeanSolid.html
4) The Catalan Solids
http://mathworld.wolfram.com/CatalanSolid.html
5) Compound Polyhedra
http://en.wikipedia.org/wiki/Polyhedral_compound
6. Tetrahedral Geometry
http://www.ac-noumea.nc/maths/amc/polyhedr/tetra_.htm
a) Every tetrahedon has a centroid.
QuickTime™ and a
TIFF (Uncompressed) decompressor
are needed to see this picture.
b) Every tetrahedron has an incenter, a circumcenter
and four excenters.
c) Every tetrahedron has a special point known as the
Monge Point which is the intersection of the six planes that
are perpendicular to a given edge and pass through the
midpoint of the opposite edge.
d) De Gua’s Theorem
QuickTime™ and a
TIFF (Uncompressed) decompressor
are needed to see this picture.
e) Bang’s Theorem
QuickTime™ and a
TIFF (Uncompressed) decompressor
are needed to see this picture.
f) Othocentric Tetrahedra
• the four altitudes are convergent (existence of an orthocentre H)
• 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