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2fca478 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 | import OAI.Geometry.PeriodicTiling.LowerDimensions
import OAI.Geometry.PeriodicTiling.ThreeDimensions
namespace OAI
namespace PeriodicTilingThree
theorem least_lattice_dimension_three :
IsLeast {d : ℕ | HasTileWithoutPeriodicComplement d} 3 := by
classical
constructor
· obtain ⟨T, hT, hTiles, hNoPeriod, _hRealTiles, _hNoRealPeriod⟩ :=
exists_three_dimensional_tile
exact ⟨T, hT, hTiles, hNoPeriod⟩
· intro d hd
by_contra hnot
have hlt : d < 3 := Nat.lt_of_not_ge hnot
obtain ⟨T, _hT, hTiles, hNoPeriod⟩ := hd
obtain ⟨A, hA, hPeriod⟩ :=
tiles_below_three_have_fullyPeriodic_complements d hlt T hTiles
exact hNoPeriod A hA hPeriod
theorem periodic_tiling_counterexample_and_minimality :
(∃ T : Finset (Lattice 3), T.Nonempty ∧
(∃ A : Set (Lattice 3), Tiles T A) ∧
(∀ A : Set (Lattice 3), Tiles T A → ¬ FullyPeriodic A) ∧
(∃ A : Set (Space 3), AETiles (Thickening T) A) ∧
(∀ A : Set (Space 3),
AETiles (Thickening T) A → ¬ EuclideanFullyPeriodic A)) ∧
IsLeast {d : ℕ | HasTileWithoutPeriodicComplement d} 3 :=
⟨exists_three_dimensional_tile, least_lattice_dimension_three⟩
end PeriodicTilingThree
end OAI
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