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Topic: Dimensions
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Murphy Waggoner

Posts: 52
Registered: 12/6/04
Posted: Jul 21, 1995 4:56 AM
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Some suggestions for helping student understand dimension.

1. Have them read Flatland: A Romance of Many Dimensions by Edwin Abbott
Abbott (Princeton University Press). This book is set in a 2-dimensional
world where no one (except the hero) believes in 1-dimensional or
3-dimensional worlds. These are descriptions of what life is like in a two
dimensional world and what objects from other dimensions world look like.
This book also happens to be a social commentary and makes for interesting
discussion (women are line segments in Flatland while men are polygons.
The more sides each man has, the higher in class he is. Therefore, women
are the lowest class. Abbott was commenting on the class distinctions of
the late 19th century (among other things).)

2. I have students consider our 3-dimensional world in the following way.

1-dimension: First I take them out to a (straight) sidewalk on our campus
running north and south. I clearly mark the four directions north, south,
east, and west (our major sidewalks and streets on campus happen to run n,
s, e, and w) on the sidewalk with masking tape. The rules are that they
can take one step north along the sidewalk (I am giving them one direction
vector), they can repeat that step as many times as they like (multiplying
the vector by a positive scalar), and they can step south as many times as
they like (multiplying the vector by a negative scalar). Then I ask them
where they can and can't go using only those rules. I try to pick the
longest sidewalk on campus so that we don't get into the walking through
building questions. 1-dimension = where you can go using one direction
(north) and its opposite(south).

2-dimensions: Now the rules are they can move one or many steps in one of
two directions (north and east) and the opposite of those two directions.
They may combine any number of steps in any of the directions in any order
they want. Where can they go now? 2-dimensions = where you can go using
two directions (north and east) and their opposites (south and west).

3-dimensions: I now ask them what they would have to do to get to top of a
tall building on campus or into a basement. How many directions (and the
opposites of those directions) do they need?

Murphy Waggoner
Department of Mathematics
Simpson College
701 North C Street
Indianola, IA 50125

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