Introduction: Pentagon Lamp
Project end result
Step 1: Inspiration
The inspiration for this project came from a lamp I saw during my visit to Freedom Lab in Amsterdam. The complex shape of the half dome and the intricacy with which all the angles matched up fascinated me. I Took a bunch of pictures and decided to build a scaled down version to fit in my kitchen.
Step 2: Geometry
Since I had space for a lamp shade with a diameter of approximately 50 cm I had to scale down the design of the lamp. Because the structure consists of a whole bunch of equilateral triangles there is a straight forward way to determine the edge length that limits the width of the lamp shade to approximately 50 cm.
The lamp shade can be decomposed in to two basic pieces. An indented pentagon "wheel" (in red) of equilateral triangles, and separate equilateral triangles (in green) to connect the "wheels". In total there are 6 pentagon wheels and 10 connector triangles.
The added screenshot of my geogebra modeling probably describes the relationships best. From left to right: The first figure is the footprint of the lamp shade. Every other edge is either part of the "pentagon wheel" or the connector triangle. I settled on a edge length of 16 cm for practical reasons and did not mind that the width was slightly larger than 50 cm. The second figure is the widest crosscut of the indented pentagon wheel. The third figure is the repeated equilateral triangle that forms the basis for both the pentagon wheels and the connector triangles. The fourth figure is a cross cut from the pentagon wheel. The fifth and final figure is the cross cut that displays the internal angle between the pentagon wheels and the connector triangles.
To get a better feel for the object I made one out of paper (template and the piece drawings will be included later because I forgot them at the workshop when I went home).
Fun Facts: My brother nicely identified the object as a partial small icosihemidodecahedron https://en.wikipedia.org/wiki/Small_icosihemidodec. Wolfram alpha also helped me in finding the exact relationship between the edge length and the circumradius. r = 0.5*(1 + (5)^1/2)*edge length => aprox 1.62*16 cm = 25.9 cm such that the diameter of the outer circumscribing circle is approx 51.78 cm.
Step 3: Prototype
I used a few prototypes to determine what width and thickness were aesthetically pleasing. I finally settled on a width of 9 mm and a 5 mm thickness. I included photos from different angles of the triangle to illustrate how the basic triangle for the pentagon "wheel" is made up.
Step 4: Sawing
I cut scrap planks of oak wood in to 5 mm thick strips. Thereafter the strips where cut to their final 9 mm width.
To handle the small pieces of wood on the table saw I made a zero clearance plate such that the pieces would not fall in to the saw. Furthermore I used a clamp to tension a stick such that it pushes the piece firmly against the fense and used push sticks and pensil/erasor to guide them through the saw.
To improve safe handling and speedup the miter cuts I bundled the sticks together using electrical tape and set a stop block for repeated cuts.
16 cm: 6 * 5 * 1 = 30
14 cm: 6 * 5 * 2 + 10 * 3 * 2 = 120
4 cm: 6 * 5 * 2 * 2 + 10 * 3 * 2 = 180
8 cm: 6 * 5 * 3 + 10 * 3 = 120
Step 5: Glue Up Edges
When all pieces where cut I glued up the individual edges. I used regular wood glue and electrical tape to "clamp" the pieces together. Afterwards I lightly sanded all pieces using a power sander. Afterwards I marked the center of the edge and two 2 cm offsets to guide in future glue ups of the triangles.
Step 6: Glue Up Triangles
Note that there are two types of triangles: The triangle that is gonna make up the pentagon wheel and the triangle that is gonna connect the wheels together.
Also make sure all pieces are glued up in the correct orientation. The second image shows one orientation mistake I made, though also came across a few isomorphism/mirror issues. Just make sure that all triangles are exact copies from each other.
Step 7: Binary Search Wheel Angle
At first I reasoned that the angle between the triangles that make up the wheel must be 108 degrees, since at each level a cross cut would result in a pentagon which have interior angles of 108 degrees. In practice this turned out to be way off. There was too few space left for the final triangle. I guess this due to the fact that the angle of the connector piece is not placed on the edge of the actual pentagon due to the dimensionality of the wooden sticks. By accident I also cut connector pieces with an angle of 160 degrees, these left way to much space after fitting the final triangle.
As I had two wrong though opposing angle evaluations I decided to empirically find the best fitting angle using a binary search strategy, dividing the search space by half and updating the upper and lower bounds accordingly. If there is too much space update the upper bound with the new angle, if there was too few space update the lower bound.
new angle = (upper bound - lower bound)/2 + lower bound.
Eventually I settled on a angle of 140 degrees.
Due to the high aspect ratio of the pieces and the weird clamping angle I used hot glue to attach them. I used a paper clip for more precise application of the hot glue.
After the prototype connector pieces I cut the final ones from the oak strips I cut earlier. I used the same tape wrapping approach to safely cut the small pieces on the miter saw. Furthermore I used a strip as backer board to prevent tear out from the saw. To prevent the small pieces from flying off once they are cut I secured them with ductape to the saw.
Step 9: Connector Type 2 to Attach Connector Triangles
The second type of connector is used to glue up the wheels with the connector triangles. The angle of the these pieces (104,6 degrees) was correctly calculated using the geogebra software as was described earlier. To attach them I glued them up in a similar hot glue fashion. Additionally I used tape to hold together the temporary structure. Later I reinforced it with additional "spokes" to give it some serious rigidity.
Step 10: Reinforcements
16 cm long spokes for internal rigidity.
Step 11: Light Fixture
Because the lamp shade is not terribly heavy I decided to just hang it from a regular electrical wire. The shade rests upon the light fixture. I added two additional zipties to ensure that the fixture stays connected to the wire in this way the weight of the shade is not transferred to two tiny screws.