Building a Dodecahedron

When we say "dodecahedron" we often mean "regular dodecahedron" (in other words all faces are the same size and shape), but it doesn't have to be - this is also a dodecahedron, even  
though all faces are not the same.

Background: You can construct a dodecahedron by cutting 12 identical, regular pentagon faces, beveling all the edges of these to half the dihedral angle and gluing them all together. The joints become the edges. Where 3 faces meet, you have the vertex. Simple enough.Now let's say you want to build a dodecahedron, but instead of boards for the faces, you want the boards to be the edges. Three edges come together at each vertex, etc.

Relative to some dimension of the paper itself, how would you determine/describe the two planes you'd cut at the end of each poster board in order to allow the three poster boards to meet and be correctly oriented in three dimensions? Let's say you want one surface of the board to split the dihedral. (i.e. boards are halfway between being coplanar with the 2 faces on either side.)

References:

Math is Fun! Spinning Dodecahedron

Matematicas Visuales

Paper Models of Polyhedra

3-D Geo Calligraphy

Tessellations, or regular divisions of the plane, are arrangements of closed shapes that completely cover the plane without overlapping and without leaving gaps. For shapes to fill the plane without overlaps or gaps, their angles, when arranged around a point, must have measures that add up to exactly 360 degrees. 

Typically, the shapes making up a tessellation are polygons or

similar regular shapes (like square tiles used on floors). Escher exploited these basic patterns in his tessellations, applying reflections, translations, and rotations to obtain a greater variety of patterns. He also “distorted” these shapes to form animals, birds, and other figures. These distortions had to obey the three, four, or six- fold symmetry of the underlying pattern in order to preserve the tessellation.

Over 2,200 years ago, ancient Greeks were decorating their homes with tessellations, making elaborate mosaics from tiny, square tiles. Early Persian and Islamic artists also created spectacular tessellating designs. More recently, the Dutch artist M. C. Escher used tessellation to create enchanting patterns of interlocking creatures, such as birds and fish.

Carver High art students have taken Escher's tessellations to the

next level by using their names as organic or geometric shapes. Through careful guidance my students have learned how to create graffiti styled fonts and fit the design within triangular shapes that join to make a beautiful geometric design.

Resources:

Tessellations project: Intro to Geometry

Tessellations Projects count as a Test Grade

Tessellation Project Rubric