Shadow Sculpture

We just had a fun, crazy and exhausting day for Super Science Saturday, a yearly K-12 outreach by our outstanding Regional Math and Science Center. 

The project had its start when my fellow traveler Heather Harrington sent me a picture of an awesome (and, later, prize winning) installation piece from Anila Quayyam Agha at ArtPrize.  (Here’s a good article about the installation.) When I saw the the RMSC was focusing on light for Super Science Saturday the bulb went on.

Intersections, by Anila Quayyam Agha

How could  that not inspire some excellent art and math?

Our handout gave a little of the background, and some basic steps to cover.

Shadow Sculpture

People really gave it a go. Mostly the kids, although we did get a higher percentage of parents trying than expected. (Those who won't join in - what keeps them back? These are folks who are giving up a Saturday for their kid to do science, so...)

Over and over we tried to emphasize:

• choose or make a structure (math)
• predict what the shadow will be
• test it out with a flashlight or on the overhead (science)
• compose an image to capture  (art)

Some of the results:




Here's a few of the final images. (You can look at all of them, too.)

We saw a lot of good mathematical and artistic problem solving. People measuring struts they'd need, considering options to construct, reverse engineering things they saw in other polytopes. They sought help when they were stuck, collaborated with neighbors and could verbalize what they were doing. They experimented with shadows, trying to figure out what angles of flashlight or position on the overhead showed the structure they wanted.

It was difficult to get people to predict. The ones who did mostly found it more interesting to do the shadows and spent longer trying to match. For some reason, this was the highest level of parental involvement.  The kids who were comparing shadows to a prediction drawing often had the parents jump in and want to see. Some people were interested enough in the connection with their predictions to trace the final

Some of what people seemed to learn about light and shadow is to start thinking of it as a projection from a single source, instead of a silhouette.  People with right rectangular prisms or right pyramids were surprised at what you could see. Kids learned about the moving of shadows with the light source, and the effects of being closer or farther from the source.

Anna's gif experiment

Artistically people divided up between looking for symmetry, being representational (lots of houses) or crazy abstract. (My favorite to see, though when building I always go for symmetry.) There was a lot of pride of ownership - the vast majority of kids wanted to take their structure with them, and they were almost always excited to bring them back to show Heather, myself or a volunteer for the light work. Heather thought it was important for people to have a chance to enter into the art, to interact with the shadows, and that's why we did the overhead projector. It was a huge hit, and there was often a line of people waiting to see theirs on the big screen. Heather's other cool idea was to capture motion, as the changing shadows are visually interesting. We experimented with gif capture and Vines, but nothing was efficient enough to be able to handle hundreds of people.

2100 straws and 1000 pipe cleaners later, we were exhausted but energized. 

We found the small aperture bar straws to be best, and picked up ours at Gordon Food Service, 500 to a box. About half a length was a good basic distance for the projector and flashlights - a whole 8" straw scale was too large. The flashlights were just $1 9 LED cheapos, but they were bright enough for strong shadows in an only half unlit room. The bulk pipe cleaner pack was the best deal, all the same color so as to avoid "I wants",  cut into 5 or 6 pieces was enough to hold two straws together. If a pipe cleaner was loose in a straw, we'd hook the end of the pipe clearner to get a better grip. Originally I was going to cut the straws into Zome like proportions, but it was unnecessary as people mostly didn't make the classic solids. Flickr seemed to be the most efficient photosharing tool so that people could get the photos afterward. Take the picture from within the Flickr app, it uploads automatically, and you can set it to a public album with one touch.

People had the option to leave their sculpture and thank goodness some did because we recycled/cannibalized everything.  

When people had the choice of tons of cool science activities, what kept us so busy? Some of those had lasers!  I think it was the art aspect. In the other activities, though cool, you were doing what someone told you to do. What we had wasn't for everyone. The option to choose a premade sculpture was good to have (we had about 10 premade sculptures), as it allowed people to still engage in the shadow prediction and art. 

Great day, good to collaborate with an artist, and a lot of creative mathematics. Who knows what mathematics dwells in the hearts of men?

Christmas Lights

aMacHan @ Flickr

Our family put up a Christmas tree this weekend, and it reminded me of one of my favorite problems ever! Students did great work on it, and developed a number of techniques.

On the weekend after Thanksgiving Karen put up a 2 meter tall Christmas tree, 170 cm wide at the base. She put an astounding 1500 lights on the tree. Write a story problem using this information.

One thing you should know about Karen's tree decorating is that she believes that lights should be distributed uniformly throughout the tree, not just on the surface. It's like the tree is full of lights. She started at the base, and put on 750 lights. Then she asked me how many will she need to finish? Now that we know she needed 1500, how far up from the ground did she get with those first 750 lights?

EDIT:
I saw that Abstruse Goose, an funny, edgy cartoon that's hip to math & physics also had some nice Christmas tree problems...

Solid Unit

As our semester in geometry drew to a close, we investigated solids.  Rather than dispense formulas (Go, Kate!), we tried to follow the Van Hiele levels.  Play and touch to get some understanding and visual recognition, sorting to start thinking about characteristics, and summarizing findings in definition-like descriptions.  Then we started thinking about measures.  Surface area is so natural, especially combined with the idea of a net.  But what about volume.  It's very interesting to have students sort the power solids by volume.  The sphere and hemisphere are very subtle.  I find it's also very common for students (college math majors) to be unable to remember formulae.  "Isn't there one with a 4/3?"

We filled them with water, and with no instructions from me, they immediately set out to try and verify their conjectured order.  (It's not as messy as you'd think.  The top of the solids container makes a pretty good tray.)  Two methods come up:  adopting a unit, and measuring each of the shapes in terms of the smallest, and filling one and trying to pour it into the next.  The different methods lead to noticing different things.  It seems like the groups that adopt a unit notice more numerical relationships from the data, and the groups that directly compare notice more of the geometric properties of the solids themselves.  ("Where does the water go?")

Usually from this data, you can suggest the idea of comparing solids with similar relationships.  Cone, Sphere, Hemisphere, Cylinder; Triangular Prism-Pyramid, Cube-Square Pyramid, Cylinder-Cone; Square Prism-Rectangular Prism-Cube, Small Triangular Prism-Large Triangular Prism or Hexagonal Prism.  Brilliantly designed little set.  (Although it does get into the experimental error that Dan Meyer cautioned about (was excited by?) in his TEDx talk.)  We compare the exterior of the solids and the water compares the interior.  Significantly different for the smallest objects.  Students have suggested measuring then by immersion, but we have yet to try it.

So this brings us to the boundary of informal and formal argument/reasoning.  How can we relate the volume of the prism and pyramid.  I do like models that fit together, but then that's just one example.  Of course, then, I tried to model it in geogebra.

Webpage or geogebra file.




It didn't help most of the students.





So I tried again:

Webpage or geogebra file.





This was helpful.  Or far more helpful, anyway.





Both sketches make use of Cavalieri's Principle to show equivalent volume.  We got at that in class by doing some block building, where each student had the same number of blocks per level.  This we extended into understanding the volume of a generalized cylinder.

Resources:  One of my favorite resources for this kind of classical problem is David Joyce's Java implementation of Euclid's Elements.  Book XII is the one you need for these problems, especially Proposition 7 and 10.  Our department's java wiz David Austin is the one who connected us to those.  David A's visualization work is literally inspiring, and worth checking out.

We finished this all by building with polydron tiles a plethora of polyhedra.  I love how, left to their own devices, students invent regular polyhedra, antiprisms, various truncations, and completely original solids.  (Unfortunately these are pretty expensive, but they are durable and usable by kids as young as 2nd grade.  Cf.  ETA Cuisinaire.) I set them the challenge of building a polyhedron with volume between 1 and 2 liters as a fancy new container for a boutique.  The need for actual measurement and estimation as well as decomposition and formula use makes this quite a challenging problem.

Extension:  as I was thinking about this and looking for resources, I came across Archimedes' proof that a sphere is 2/3 of the circumscribed cylinder.  Famously, this is the relationship that Archimedes wanted put on his tomb.  I took the translation from the Archimedes' Palimpsest that was posted at Cut the Knot (an invaluable geometry site), added some clarifying comments and made it into a handout with an accompanying geogebra sketch.  The sketch isn't really for visualization, but allows the reader to experimentally test some of Archimedes' unjustified claims.  (All correct, though.  Were it today, the justification of the steps would be left as an exercise for the reader.  Pretty good exercise.)

Quick Triangle Sum & Pythagorean Proof

Just a quick, pretty unoriginal sketch to help secondary/tertiary students think through the justification for the sum of the angles in a triangle.

As a dynamic webpage or the original geogebra file.

EDIT:  As my students investigated (see these sketches elsewhere), they got interested in proofs of the Pythagorean Theorem.  Since they were investigating so nicely on their own, with interesting results, I had time to draw up a familiar proof in Geogebra.  Webpage or geogebra file.  The webpage has some additional hints to help towards a proof.

Net Result

I really enjoy designing nets (2-D plans that fold up into 3-D objects), and I love designing them in dynamic geometry, where you can design all nets. If I ever got the time to do math research again, I can see going in that direction.

I designed these four sketches for my geometry class, which is working on a project to design their own package with a few constraints. Each sketch is available as a dynamic webpage or the geogebra file. Here's 5 nets that were made with the sketches, in a printable pdf format. Geogebra actually has very nice priniting controls, so if you're interested in designing your own, choose that option. Remember you can install it, or run it from your browser at geogebra.org.

General tetrahedron: webpage or geogebra file

Square pyramid: webpage or geogebra file

Convex oblique pentagonal prism: webpage or geogebra file

I was very disappointed that the above sketch, though intuitive, couldn't make concave prisms. This next sketch is the answer, but would be a muddle to try to figure out how it was made. The net is pretty though, for designing solids, and I'm proud of it as work.

General Oblique Pentagonal Prism: webpage or geogebra file

Note that you can use the pentagonal prism nets to make quadrilateral and triangular prism nets by making some of the base vertices collinear.

Let me know what you think, and send me your dynamic geometry design challenges!

PS> I also wrote up a memoir for my class of how I made the pyramids (which really sounds egotistical; reminds me of a Tom Lehrer line about "even the Pharoahs, had to import, Hebrew braseros" Listen at the link. If you do, check out Lobachevsky, a great math song.) Sorry for the ramble - I'm tired. Here's the memoir.