I've currently moved over to blogging at http://mathcraft.wonderhowto.com/

Here I will be posting about 4 times per week and will have a project every week with more in depth information and a how to component and templates. There will also be a user forum for people to submit pictures and ideas.

Thanks! I hope to see you there!

Standardized Testing And Reporting

It's the middle of April and this next week all students at University Preparatory School, where I teach, will be taking the California standardized tests. So I decided to design and build a sculpture that I'm calling "Standardized Testing And Reporting" or "STAR", which is what the California testing program is named. The sculpture is made up of 80 pencils and is held together with a variety of glues.



From the side:



Two of the five identical pieces it is made from:


The sculpture is made from 5 pieces, each of which is part of a hyperbolic paraboloid embedded in a regular tetrahedron. This works because the dihedral angle (angle between faces) of a tetrahedron is 70.5 degrees which when multiplied by 5 (the number of pieces) very nearly results in 360 degrees.

The first inspiration for this work was George Hart's 72 Pencils. This gave me the idea for using Pencils. Here's my recreation of that work:


The second was Carlo H. Séquin's Ribbed Hemicube. This brought the realization that you could easily embed hyperbolic paraboloids into regular tetrahedra. Here's my recreation of that work:


The third was Erik Demaine's Polyhedra built from Hypars (Sections of Hyperbolic Paraboloids). This inspired me to create a full polyhedral structure from sections. (Note: While this looks nearly identical to what I built there are major advantages to working with these folded paper structures. You can make the angles most anything you want. For instance a 6 pointed star would work just as easily) I haven't actually completed one of his works. I'm working on it but I'm a slow folder. Here's a picture from his site:



Here's a few pictures from the build process:


Marking up the pencils so they can be glued in the proper positions:


Getting the Tetrahedral frame together:


The completed tetrahedral frame (The front pencil will be removed):


Starting the hyperbolic paraboloid by connecting pencils across the marks:


One done! (With overlap):


From the side:


Three together (testing the fit)


Five together (Still need to trim off the extra length of the pencils):


After trimming:


The completed sculpture:



7 comments:

Kelsey said...

That's awesome! And I love your explanation of how you got the idea.

cmmelis said...

This is a great Math Club project. What did you use to trim the pencils?

Cory said...

I used a dremel with a cutoff disc. Works pretty well. You have to be careful and a couple will probably get detached as a result and have to be re-glued.

The only difficulty I ever had was attachment for the pencils. The paint on the pencils doesn't seem to work with very many adhesives...superglue works but not well. The only thing that really worked well was hotglue but it's pretty messy. If I were to do it again I would spend a little while trying to find a better attachment method.

Anonymous said...

I bet small glue dots would be a great hot glue substitute.

Anonymous said...

I think I am going to get metal rods and build this out of metal! Way cool!

Anonymous said...

Can someone please walk me through the steps of making this? It is an extra credit project and it looks really complicated.

Anonymous said...

Don't like the Dixon pencil structure. I like the Dixon
pencil, but not the idea of a structure with them.
Blackwing pencils are cool.

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