Attribute decks - an alternative representation of geometric structure
An attribute deck is a set of cards, each of which is uniquely defined by i attributes, each of
which has ni values. When the ni are a constant integer (for all values of i), such decks are
models of finite vector spaces – and a little bit more.
For example, A two-attribute deck is also an ideal hands-on representation of a finite plane.
Consider an attribute deck constructed by color—red, green blue, yellow and shape—triangle,
square, star, oval). The 16 cards represent points of the plane as shown in figure 2.
Shape
Color
The cards can now be grouped as lines in the plane which may be described in terms of the
attributes. Horizontal lines are all of the same shape, but no two cards have the same color.
Vertical lines are all of the same color, but not two cards have the same shape. Diagonal lines
(slope 1 or -1, with "wrap-around" assumed) are the sets of cards in which no two cards have the
same shape or the same color. Essentially, any 4 cards make a "line" in the diagram if the four
cards either all have the same color or represent all colors AND all have the same shape or
represent all shapes.
If we can represent concepts of the coordinate plane, then we can extend the representation to
higher dimensions. For three attributes, we can model three dimensions; for four dimensions we
can model four dimensions, and so on. Consider a three attribute deck by adding the attribute of
number (one, two, three, or four) shapes on each card of the deck shown above. Now we have
"levels" depending on the value of the number attribute. As before, any 4 cards represent a line
in the geometry if all cards are of the same color OR represent all numbers AND all cards are of
the same shape or represent all shapes OR all cards are of the same number or represent all
numbers. We can also now describe planes as those collections of 16 cards which either all share
a common attribute (for example, all 16 red cards) OR no two cards in the set which have more
than one attribute in common.
Consider a deck of 81 attribute cards as follows.
ATTRIBUTES:
VALUES
Shape
circle, triangle, square
Number
1, 2, 3
Color
3 different colors
Shading
open, striped, solid
Recall the game rules for TRIPLE:
A TRIPLE consists of 3 cards with the property that for each value (independently) the
cards are all alike in that value, or are all different.
1. How many TRIPLEs are there in the entire deck?
2. What is the maximum number of TRIPLEs in any layout of 9 cards? 12 cards?
3. What is the probability that a particular TRIPLE is in a layout of 9 cards? 12 cards?
4. What is the expected number of TRIPLEs in a layout of 9 cards? 12 cards? (Hint: the
expected value is the number of possible TRIPLEs times the probability of any particular
TRIPLE.