Week 2 – Reading Summary & Activity
My reading for this week is approximately half of Johannes
Kepler’s The Six Cornered Snowflake, written in 1611 as a New Year’s
gift to his benefactor Lord Matthaus Wacker von Wackenfels. I really enjoyed this reading, and have actually
ordered myself a copy of the book! As an
ex-research ecologist and lover of both mathematics and the natural world, the
content is right up my alley!
The text has a conversational tone, where Kepler first
laments that he has nothing to give his benefactor, but then upon observing the
mathematical beauty of the snowflakes falling on his coat decides to gift him
his musings on the geometrical similarities shared by a number of objects including
snowflakes, cells of honeycomb, and pomegranate seeds.
STOP#1 – Life in the 1600s
Reading Kepler’s observations made me pause and think about what life would have been like over 400 years ago, and how exceptional Kepler would have been compared to most other people alive in Europe at the time. I’m familiar with Kepler’s major accomplishments/contributions to science – he developed mathematical equations describing the elliptical motion of planets around the sun that are still taught in science classrooms today. It’s inspiring to think that he not only spent his time and brainpower looking up at the vastness of the universe but also found interest in the small simple things around him like snowflakes, insects, and fruit. But I also think about how his life would have been radically different than the typical folk alive in northern Europe at the time – I doubt many of them would have access to pomegranates or would have the free time to contemplate the shape of their seeds.
Kepler wonders why snowflakes have 6-sided symmetry, which leads him to discuss the hexagonal structure of cells of honey comb – a shape which results since it is the most space-efficient packing arrangement. He also notes that pomegranate seeds also have a peculiar geometry due to optimal packing – the fleshy, juicy material surrounding pomegranate seeds starts out spherical but as they grow/swell as they mature inside the fruit they become more constrained and take on an angular 3D shape that is roughly a rhombic dodecahedron.
He then goes on to discuss more conceptual optimal packing scenarios
involving spheres, making observations such as how if they are packed in a grid-like
formation (with 8 neighbors in the same plane) and inflated like the
pomegranate seeds they would form rectangular prisms or spheres, but if they
are packed in a hexagonal formation (with 6 neighbors) when inflated they would
form hexagonal prisms and other 3D shapes depending on the arrangement of the layers
above and below.
STOP#2 – Words vs. Shapes
In the preceding paragraph, I described two different ways in which spheres might be packed together. I suspect that some people are able to read a passage like this and be able to visualize the situation in their mind with relative ease. Others might re-read the passage a number of times and give up in frustration, unable to “see” what is being described. Perhaps drawings or diagrams would help people in this category. I suspect there would also be others where even the diagrams might not be enough, and would require physical items to touch and hold. I wonder if someone with blindness would be better at visualizing shapes in their head than a person with typical sight, or if they would find physical objects that could be held and touched more useful?
As I was folding some of the origami shapes as part of my activity for this week, I noticed dodecahedron required 12 pentagons, and folding a 4-sided piece of paper into a 5-sided shape without any measurements was intriguing. Also, the folding instructions specified A5 paper. Much of the world uses “A” series paper but (tragically) in Canada we’ve gone with what the USA uses instead. If you don’t know, “A” series paper has some cool mathematical properties – the ratio between the two side lengths is 1:√2 (1:1.4142...) which creates a rectangular shape where by folding/cutting it in half results in two smaller pieces that both still have that same side ratio! So for example from a piece of A4 (close to our letter size) you get 2 pieces of A5.
https://www.col-print.co.uk/blog/using-a-series-paper-sizes
Coincidentally, a few days ago I received in the mail some instrument strings that I ordered from the UK. I noticed immediately that the receipt enclosed
in the envelope was a piece of A4 (not letter) paper. I planned on keeping it to show to my math
classes at some point, and here it is:
You can see the white A4 piece is
slightly narrower and longer than yellow North American letter size. Anyways, I decided to fold both of them
according to the directions to see if they both would work, as I suspected the
letter size may not quite work, but I couldn’t visualize exactly why not – time
for some embodied mathematics!
Turns out my hunch was correct – the folded pentagons revealed that the A4 paper was closer to a regular (equal-sided) pentagon. The letter paper (yellow) had two shorter sides (the “walls” if you imagine it as an outline of a house). It might be difficult to tell from the photo, but the shorter sides were almost a centimeter shorter than the other three.
STOP#3 – Embodied Learning
I find I’ve always been able to visualize things in my mind with accuracy and ease, and enjoy the challenge and imaginative aspects of doing so, so I found having the physical items like the pomegranate, honeycomb, and platonic solids were not helpful or necessary for me. I was however pleased to encounter the situation above where I could not think my way through the problem and needed to fold the two different pentagons to see the result!
Another interesting post from you, Reed. Pleasure to read and very engaging. I can totally relate with when you describe that some people are able to read and visualize things in their head while others have to keep rereading as they are not able to visualize what is being described. I definitely fit into the second category. When it comes to geometry, measurements and describing shapes, I defnitely feel that having a 2-d drawing or 3-d model helps visualise things better.
ReplyDeleteFor the same reason, I love assembling ikea furniture as I find it very easy to follow the the drawn instructions, my sister, on the other hands finds it very frustrating due to lack of written instructions. There might be some people who would actually feel more comfortable in having an actual 3-d model in front of them before they start assembling.
And it all loops back to the same point that in our teaching - to be able to cater to all students, we need to make sure that we are offering multifaceted and engaging ways to offer equal access to learning for al students.
Ikea furniture is a great example! I also like the visual/drawn instructions they provide, but like your sister every once in a while I wish they made something a bit clearer using language to describe a detail that visually isn't that clear. Like you say, offering more learning options leads to more equal access to learning (and more success) for all.
DeleteI'm always in awe of those who can visualize objects and how they fit together in their head. My husband has that ability and he uses it for everything from playing tetris, to packing luggage in the car, to building furniture or putting together an engine. I on the other hand am seriously lacking in the spatial reasoning domain. At the very least I require a diagram, but I would prefer to be able to hold the shapes and see how they fit together from all sides.
ReplyDeleteI've noticed in my classroom, that the majority of students struggle to interpret and create 3D diagrams, yet when they have the physical objects in front of them they are better able to understand and describe the attributes of the shapes. Obviously these observations are formed predominantly through the sense of sight, but I think it would be really interesting to have them interpret the shapes through touch instead.
I wonder if they would explore the shapes with both hands in a symmetrical manner and work their way in from the outside, as described in the study that Kanwaljit read this week? I also wonder if they would demonstrate an improved understanding of the 3D diagrams in relation to the volume and surface area etc?