Solution 28: Configurations.

Index

28. (a) We know that b = v and that m = n = 2. So each point is incident with two lines, and each line is incident with two points, and the number of lines is the same as the number of points. The only geometry that is possible when b > 3 is the "b-gon geometry" (see below where b = 5 ).

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When b = 2 no b2 configuration is possible.

(b) The trick here is not to neglect the third column of the figure below where the representation of the model is by points and non-incident relationships.

The reason is that the figures are much simpler this way! Certainly the models in the first colunmn above and the column below all have 9 points and 9 lines each, and that each point is incident with 3 lines and each line is incident with 3 points. That is, b = v = 9, and m = n = 3 for each of these models. But that doesn't prove that any one of these models is isomorphic to another. First, note that the third column above shows that these three models are distinct: no one is isomorphic to another. This is because they can't possibly have the same incidence relationships. To show then that the first model below is isomorphic to the first model (in row 1) above, that the second model below is isomorphic to the second model above, and that the remaining two models are isomphic, all we need to do is to draw the non-incidence relationships for the models below to see that they match those from above.

This give us the following models, which are isomorphic to those of column three in our original figure.