# Patterns with Inkscape, then extruding 2D to 3D

## Introduction

Patterns are all around us. We can observe them in our homes, schools, where we work, in our art, and in our architecture. A repeating pattern without gaps or overlaps is known as a tessellation. In a tessellation there is a base tile that is repeated over a plane.

There are 17 known symmetry groups that form what is referred to as wallpaper patterns. These symmetry groups refer to two-dimensional repetitive patterns and have been categorized by their symmetries: the types of rotations, reflections, translations, and glide-reflections.

• Rotations are clockwise and can rotate by half-turns, 120 degree turns, 90 degree turns, and 60 degree turns.
• A reflection is a flip along an axis:horizontal, vertical, or at some angle.
• A translation is a move in which everything is moved by the same amount in the same direction.
• Glide reflections are composed of a reflection across an axis and a translation along the axis.

It may seem surprising that there are only 17 known patterns for tilings, but Mathematician Evgraf Fedorov provided a proof in 1891 and George Plya provided another for this fact in 1924. A chart created by Dorothy Washburn and Donald Crowe can help you determine which of the 17 patterns your tiling is part of:

### Parsing the names

• Groups start with either p or c (primitive cell or centered cell).
• The number n represents the type of rotation.
• m refers to mirror.
• g refers to glide.
• 1 refers to translation.

So for example Symmetry group 1: p1  is made up of only translations. The picture remains unchanged no matter how many translations you apply. This group has the simplest pattern and is easy to see:

Symmetry group 2: p2  is made up of both translations and rotations. 180 degree rotations are referred to as half-turns.

Symmetry group 3: pm has reflections and translations. There are 2 types of parallel reflections axes. These reflections are referred to as bilateral symmetries.

Symmetry group 4: pg contains glide reflections and translations. Glide reflections are sometimes difficult to find when you are looking at the pattern. The direction of a glide reflection is parallel to one axis of translation and perpendicular to the other axis of translation.

Symmetry group 5: cm  contains reflections and glide-reflections with parallel axes and translations.

Symmetry group 6: pmm contains reflections whose axes are perpendicular. The rotations are half-turns.

Symmetry group 7: pmg has reflections and glide reflections, as well as translations. The red lines are the axes of reflection, green lines are the axes of glide reflections, and black dots are the fixed points of the half-turns.

Symmetry group 8: pgg  contains glide-reflections, half turn rotations, and translations.

Symmetry group 9: cmm  contains reflections and 180 degree rotations.

Symmetry Group 10: p4  contains rotations and translations. This group contains 90 degree rotations,

while the previous groups do not. The black dots in figure 11 represent the centers for the half turns and the blue squares represent the centers of the 90 degree turns.

Symmetry Group 11: p4m  contains rotations, translations, and reflections. The rotation centers lie on the reflection axes. There are also glide-reflections in this group. This group contains 90 degree turns, half turns, and reflection axes at 45 degree angle.

Symmetry Group 12: p4g This group contains reflections, glide reflections, and rotations. The axes of reflection are perpendicular at 90 and 180 degree angles. The lattice is square.

Symmetry Group 13: p3 This group contains rotations and translations. The lattice is a hexagon. The rotations centers are found at the vertices and centers of the triangles, and are 120 degree rotations.

Symmetry group 14: p31m This group contains reflections, rotations, and glide reflections. The reflections are at 60 degree inclines to each other and the rotations are 120 degree turns. The lattice is a hexagon. The reflection axes are parallel and make equilateral triangles. The axes of glide reflections are halfway between the reflection axes.

Symmetry Group 15: p3m1 This group is similar to p31m. The difference is that all of the centers of rotation lie on the reflection axes. The lattice is a hexagon.

Symmetry Group 16: p6 This group contains rotations and translations. The rotations are at 60 degrees, 180 degrees, and half turns. The lattice is a hexagon. There are no reflections in this group.

Symmetry group 17: p6m This is the most complicated group of them all. It contains reflections, rotations, translations, and glide reflections. The rotations are at 120 degrees, 60 degrees, and 180 degrees.

The lattice is a hexagon. The axes of glide reflections are halfway between the parallel reflection axes. They pass through the centers of the half turns.

There are three regular polygons that satisfy the requirement of covering a plane without gaps or overlaps. These regular polygons are:

• Triangles
• Squares
• Hexagons

There is a way to symbolize the tilings using numbers. Each number in the sequence denotes the polygon at each vertex.

• 333333 is the symbol for the tessellation of triangles.
• 4444 is the symbol for squares.
• 666 is the symbol for hexagons.

A complete set of tiling symmetries requires taking these shapes and adding rotations and reflections. Remember there are only 17 known tiling symmetries.

## Making a Circular Pattern with Inkscape

1. Open Inkscape.
3. Double click on the fill stroke area in the lower left area to open the fill and stroke window:
4. Click on fill and the x to have no fill:
5. Click on the stroke style tab and make the width at least 5 pixels:

6. Select Edit:Clone:Create Tiled Clones:

7. For Symmetry, select P1 and set Rows to 1 and Columns to 6:

8. Click on the Shift Tab and enable Exclude Tile option:

9. Click on the Rotation tab and change the Angle in per column to 60°:

10. Click on the Selection Tool and press SHIFT+S for Make Smooth:

11. Drag the center point down to change the Center of Rotation:

12. Click on the Create button:

13. Click on the starting tile. You will need to make this shape a path:

14. Select Objectbjects to Marker:

15. Click on the Edit path tool:

16. Click on the Convert button:

17. Make adjustments so that you have a continuous path:

18. Continue:

19. Avoid shapes that are too small:

21. When you are done select the Selection Tool and press CTRL+A to select all the elements. Press CTRL+C to copy them:

22. Create a new document by pressing CTRL+N:
23. Press CTRL+V to paste the copied elements.
24. Click on Paths:Trace Bitmap:

25. Click on Update:

26. Click OK and close the window:

27. Click on the Edit Path tool to test that you really have a path:

28. Save the Image as an Inkscape SVG to use with TinkerCad or as a PNG to use with Creator:

1. Open TinkerCad in a webGL enabled browser like Chrome.
3. Create a New Design.
4. Click on Import to open window:
5. Click on Choose File:

7. Set your parameters to 10% and 1mm:

8. Click on Designroperties to rename:

9. Rename and save changes:

10. Click Designownload for 3D printing:

11. Select STL:

12. Print.

## Extruding with Shapeways' 2D to 3D Creator

1. Open Shapeways' 2D to 3D Creator in a browser.
2. Click on the Choose file button:

3. You can adjust the width and thickness parameters:

5. If you have a version of Netfabb Basic pre-October 2016, open it.

6. CTRL+click on the model and export it as an STL file:

7. When you save the file, add the STL extension:

8. Print.