The Set of Sets in SET
The Set of Sets in SET
BitXor gives the bitwise XOR of a set of integers in binary form. This uses the finite field ={0,1}. Is there something similar for ?
F
2
F
3
The ternary representations of are ; transposing,
=
, which gives . Similarly, in ternary is with transpose . In each transpose, the triples of numbers are either all the same or all different. Under a ternary form of BitXor, the triples have a ternary bitwise sum of zero.
(31,38,51)
(1011,1102,1220)
1 | 0 | 1 | 1 |
1 | 1 | 0 | 2 |
1 | 2 | 2 | 0 |
1 | 1 | 1 |
0 | 1 | 2 |
1 | 0 | 2 |
1 | 2 | 0 |
(111,012,102,120)
(45,48,51)
(1200,1210,1220)
(111,222,012,000)
In the game of SET, there are 81 cards, each having four features: number of shapes (one, two or three), shape (diamond, squiggle or oval), shading (solid, striped or open) and color (red, green or purple). A set of three cards is called a "set" if the four features are pairwise identical or completely different. This is the same as a bitwise sum of zero. SET can be thought of as projective geometry .
F
3
PG(3,4)
There are 1080 possible sets in SET. This Demonstration shows them all, using 40 decks of SET to show 27 sets at a time. This is also a Steiner system: each pair of numbers from 0 to 80 is represented in a unique set. This is also a resolvable design.
81–(27,3,1)