Introduction

A tessellation, is a tiling of the plane with no gaps or overlaps. The most common tessellations today are floor tilings, using square, rectangular, hexagonal, or other shapes of ceramic tile, but other shapes can be tessellated.

Translation

Take a nibble from the top and translate it down.
Take a nibble from the left and translate it to the right.

Rotation

Take a nibble from the top (constrain the nibble to the top half) and translate and rotate it CCW so that it fits along the right side.
Take a nibble from the left and translate and rotate it CCW so that it fits along the bottom.

More challenging rotation

Start with a hexagon.
Take a nibble from a side and translate and rotate it CCW so that it fits along the edge to the right.
Continue until all sides are covered
There is not one right way to do this project. It can be accomplished using a combination of commands. The goal is for you to break down the problem and use what might be the best series of commands to acheive your goal.

Once you have decided on the type of tile to construct and have created it, you'll need to create an offset. Why do you need an offset? A cylinder will not fit into a hole if they both have the same diameter. If you want a cylinder to fit snuggly into a hole make the diameter of the hole .4mm larger. If you want the cylinder to slide, make the hole at least .8mm larger. The offsets are known as tolerances and are determined by your printer. In order to create an offset, you'll need to have a continuous line or polyline.

Designing the Tile

1. If you don't have Inkscape, download and install it from here. Otherwise open the application.
2. Create a New Document. CTRL+N
3. Select the Rect tool (F4), hold down CTRL and drag a perfect square.
4. Take a nibble from the top with the beziér tool. Make sure to come out at one of the corners:
   
5. Select nibble and square and select Path>Division
   
   

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6. With the Select and Transform Object tool, just select the nibble. Move to the left. Select Object>Rotate 90° CW
   
7. Fit the shape against the left edge.
   
8. Make sure there is no stroke.
   
9. Select the nibble and the square and then select Path>Union
   
10. With the Beziér tool, create a nibble from the bottom. Make sure you come out at a corner
    
11. Select nibble and square and select Path>Division
12. With the Select and Transform Object tool, just select the nibble. Move the nibble to the right
    
13. Select Object>Rotate 90° CW
    
14. Fit the shape against the right edge. Make sure there is no stroke.
    
15. Select the nibble and the square and then select Path>Union
    
16. Change the measurement to millimeters
    
17. Resize the tile to have a width of about 80mm. Lock the width.
    
18. Set the stroke to .5mm
    
    
    
    

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19. Create a duplicate of the tile by pressing Edit>Duplicate
    
    
    

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20. Change the fill color of the duplicate.
    
    

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21. Turn off the stroke
    
    
    

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22. Delete the original tile.
    
    

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23. You want to resize the document to your image. Press File>Document Properties, toggle open Resize page to content and click Resize page to drawing or selection
    
    
    
    

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24. Save your tile as an SVG file
    
    
    

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25. Import the SVG into Tinkercad and extrude.
    

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Fusion 360

1. Create a New Sketch:
   
2. Select the ground plane:
   
3. Make a centered rectangle 80 x 80:
   
4. Select the spline tool:
   
5. Make a nibble at the top:
   
6. Make a nibble on the left side:
   
7. Stop the sketch:
   
8. Extrude one of the nibbles:
   
9. Turn on Sketch 1:
   
10. Extrude the other two parts, making sure to create new bodies with each extrusion:
    
11. Select the top nibble:
    
12. Move it 80mm to the bottom:
    
    
13. Select the nibble on the left and move it 80mm to the right:
    
14. Select Combine from the modify menu:
    
15. Select all three pieces and click OK:
    
    
16. Create a sketch on the top surface:
    
17. Create an offset of -.4mm:
    
18. Turn off the body and Sketch 1:
    
19. Extrude the new profile. Make a copy to test fit:
    

Extension

Design a box in the same shape as your tile to hold the tiles.