Section2.6The Projective Planes from Affine Spaces
Subsection2.6.1Back to the 3d
Activity2.6.1.
We now return to our affine three space.
(a)
Reconstruct the 3d space with SET cards with the caveat that the middle most card must be Two Green Striped Diamonds.
(b)
Take each of the SETs in the 3d space that contain Two Green Striped Diamonds and find the corresponding \(\mathbb{P}\) cards. How many of them are there.
(c)
Try to form these in the same configuration as in Activity 2.1.6, how close can you get?
(d)
What connection can you see between the lines of the projective plane you are building and the 3d affine SET space?
Activity2.6.2.
Let’s go the other direction, reproduce the exact plane you originally built in Activity 2.1.6.
(a)
Pull out all the SET cards that appear on the \(\mathbb{P}\)SET cards and add to them Two Green Striped Diamonds.
(b)
Try to use these cards to fill out a 3d-affine SET space.
Problem2.6.1.
Based on what you have observed, carefully capture what you think is happening with a conjecture.
If we replace Two Green Striped Diamonds with some other card, what, if anything, what if anything stays the same? Try to prove any claims you make.
Problem2.6.4.
If we replace the 3d affine SET space with the standard \(\mathbb{R}^3\text{,}\) and replace Two Green Striped Diamonds with the origin \((0,0,0)\text{,}\) how would your conjecture you found in Problem 2.6.1 be adjusted?
Do you think this new version of the conjecture should be true? Explain your thinking. Can you prove any claims?
