1. Quasi.py An Investigation into Tessellating Three Space with Quasicrystals

2. Quasiperiodic Tiling  Discovered by Penrose in 1974  Used only two shapes: fat and skinny rhombi  Original rule for generation was do not arrange such that two rhombi form one big rhombus  Discovered by Penrose in 1974  Used only two shapes: fat and skinny rhombi  Original rule for generation was do not arrange such that two rhombi form one big rhombus

3. Quasiperiodic Tiling

5. Ammann Bars  Discovered simultaeously by Penrose and Ammann  New method for generating 2D quasicrystals  Crystals should be placed such that bars become continuous lines  Not foolproof  Discovered simultaeously by Penrose and Ammann  New method for generating 2D quasicrystals  Crystals should be placed such that bars become continuous lines  Not foolproof

6. Ammann Bars

7. DeBruijn’s Two Methods  In 1980 Nicolaas DeBruijn came up with two foolproof methods for generating quasicrystals  These methods have applications in both two and three dimensions  Projection method  Dual method  In 1980 Nicolaas DeBruijn came up with two foolproof methods for generating quasicrystals  These methods have applications in both two and three dimensions  Projection method  Dual method

8. Projection Method  Start with a lattice of higher dimensional cubes (5D for a 2D tiling, 6D for a 3D tiling)  Slice this lattice with a plane with slope equal to the golden ratio (1:(1+√5)/2)  Construct a new plane perpendicular to first plane  Project points onto new plane to see if they fit inside the gate  If they fit, join the vertices with line segments of equal length  Start with a lattice of higher dimensional cubes (5D for a 2D tiling, 6D for a 3D tiling)  Slice this lattice with a plane with slope equal to the golden ratio (1:(1+√5)/2)  Construct a new plane perpendicular to first plane  Project points onto new plane to see if they fit inside the gate  If they fit, join the vertices with line segments of equal length

9. Ramifications of Projection Method  Quasicrystals occur in nature (Al- Mn crystals, Zn-Cr crystals)  If the projection method describes how these crystals form, then that provides evidence of existence for higher spatial dimensions.  Quasicrystals occur in nature (Al- Mn crystals, Zn-Cr crystals)  If the projection method describes how these crystals form, then that provides evidence of existence for higher spatial dimensions.

10. Dual Method  Start with a star of vectors normal to the faces of a dodecahedron  Draw planes normal to those vectors at unit distances away from the origin  At points where three planes intersect, transport corresponding vectors and perform the vector addition, drawing a rhombohedron, either fat or skinny  Drawing enough rhombohedra reveals a quasiperiodic pattern  This is the method I employed in my program  Start with a star of vectors normal to the faces of a dodecahedron  Draw planes normal to those vectors at unit distances away from the origin  At points where three planes intersect, transport corresponding vectors and perform the vector addition, drawing a rhombohedron, either fat or skinny  Drawing enough rhombohedra reveals a quasiperiodic pattern  This is the method I employed in my program

11. On to the program!

12. Program Goals  Take six vectors and use them to calculate the positions of quasicrystals  Correctly chose which quasicrystals should be drawn and which are not worthy  Take six vectors and use them to calculate the positions of quasicrystals  Correctly chose which quasicrystals should be drawn and which are not worthy

13. calculatePosition()  Takes three vectors and a material node as arguments  For each vector it generates an equation of a normal plane  Calculates point of intersection, if it exists, using cramer’s rule  Passes the vectors, the point of intersection, and the material node to the appropriate drawing function  Takes three vectors and a material node as arguments  For each vector it generates an equation of a normal plane  Calculates point of intersection, if it exists, using cramer’s rule  Passes the vectors, the point of intersection, and the material node to the appropriate drawing function

14. drawQuasiCrystal()  Takes a point, three vectors, and a material node as arguments  Creates a points node, several index nodes, and drawable nodes that are line strips to draw a wire frame quasicrystal  Takes a point, three vectors, and a material node as arguments  Creates a points node, several index nodes, and drawable nodes that are line strips to draw a wire frame quasicrystal

15. Issues with quasi.py  Crystal selection sucks  It would be nice to draw the rhombohedra one at a time  It would be nice to draw some rhombohedra as solid  Crystal selection sucks  It would be nice to draw the rhombohedra one at a time  It would be nice to draw some rhombohedra as solid

16. Screenshots