Lost in 4-space.
Richard BarwellAmongst the many thoughts, insights and ideas, I took away from GCEDM/CMESG 2000, there was some mathematics, stimulated by the topic session offered by Vicki Zack and David Reid. In their session they talked about some work they have been doing with Vicki’s class on sums of n and sums of n squared. Actually, I do not remember the discussion too clearly, since I became absorbed by an image which David showed us, which he had used with Vicki's class.
University of Bristol, Graduate School of Education.
The image, which was three dimensional, was related to sums of squares. I also had a new challenge - to think about what happens in four dimensions.
What does a pyramid look like in 4D? How can it be represented? How many
are needed to fit together? How do 4D pyramids fit together? These and
related questions not only absorbed me for the rest of Vicki and David’s
session, but for the rest of the conference and the several months which
have followed. At first I found it difficult to work with the extra dimension.
During the breaks I thought and I talked with anyone who was interested
and I sat around with glazed eyes, lost in 4-space, my brain aching.
Later in the conference, instead of attending a plenary discussion,
I chose to go to the coffee room where there was a crate of multilink cubes.
I prepared sets of cubes, as representations of 13 + 23
+ 33 + 43. Another way of thinking about this
is as four slices through one dimension of a 4-pyramid, in the same way
that slices through a 3-pyramid produces increasing squares. A small group
of us sat around seeking a way to fit them together, as well as thinking
what fitting-together is like in 4-space. I got left behind, particularly
at a point where the group had a moment of insight which I could not see,
whereupon things moved very fast. The result was four cuboids pieced together
from parts of four 4-pyramids. Another image. An image of images, in fact.
We looked at the cuboids, remarking some of the patterns which move through
and across from one cuboid to another, starting to make sense of them.
It feels important that I was still able to do this, despite having been
left behind during the construction of the images. Later I sat and produced
layer diagrams for each cuboid, side-by-side in my notebook. (See Diagram
2) I started to see more patterns and make more sense of where the
pyramids were. I realised that cuboids can be sliced in more than one way
and produced a second set of slices which I could compare with the first,
allowing me to see more general patterns.
What can I say about this process of seeing more? Although the original
motivation of putting pyramids together was to think about formulas for
sums of n or n-squared or n-cubed, for me there was also the mathematics
of increasing dimensions, a process of increasing generalisation and abstraction.
In this, the patterns seem to be important: my access to the mathematics
is made possible by the rich patterns found in these images. I am also
aware that the process of seeing more is a joint activity. What I have
described here came about through talking, thinking and seeing with others.
At different moments, different participants had different insights which
moved all of us forward. We shared various images, yet transformed them,
and so ourselves, through our talk, so that colours and shapes became patterns
and then patterns of patterns.
And so the process continues. I carried the notebook and my images home
to the UK. I am still absorbed by them - there is so much that can be seen
in them. I have reconstructed the cuboids using multilink cubes, gradually
making more sense of what is happening, at some stage moving beyond the
point where I was originally left behind in the coffee room. I have started
to extend the ideas to 5 and 6 dimensions, creating more complex patterns,
through which yet more general patterns emerge. Josee, my wife has, grown
used to heaps of multilink cubes all over the place, and to me staring
at them, lost in space.
He used multilink cubes to show 12 + 22 + 32
+ 42, in the form of a ‘pyramid’ consisting of a 4x4 square
layer, then a 3x3 layer, a 2x2 layer and a 1x1 layer. 
Using
three such pyramids, David fitted them all together to make something that
was almost - but not quite - a cuboid. We thought about how this
image can be used to develop the formula forSn2.
I became fascinated and began to wonder if the same approach could be extended
to finding the formula for Sn3,
a move which requires four dimensions. First, I needed to understand how
the three pyramids so magically fitted together. To work on this, I needed
a way of examining the whole structure of the nearly-cuboid, rather than
just looking at the exterior. I therefore sketched each layer of the solid,
represented as five rectangular 4x5 grids, with each square shaded according
to the colour of the cubes. It is this activity of visualising how the
three pyramids fit together in order to produce the layer representation
that absorbed me for much of the rest of the session. I now had a new image.
[See diagram 1]
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