FormalConjectures.ErdosProblems.«633»

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import FormalConjecturesUtil

Erdős Problem 633

Reference:

erdosproblems.com/633

[So09] Soifer, Alexander, How Does One Cut a Triangle? I

[So09c] Soifer, Alexander, Is there anything beyond the solution?

open Affineopen scoped Congruent EuclideanGeometry Similarnamespace Erdos633variable {n : ℕ} {T : Triangle ℝ ℝ²}variable (n T) in

A triangle is n-cuttable if it can be decomposed into n congruent triangles.

def IsCuttable : Prop :=
  ∃ Ts : Fin n → Triangle ℝ ℝ²,
    (∀ i j, (Ts i).points ≅ (Ts j).points) ∧
      Pairwise (fun i j ↦ Disjoint (Ts i).interior (Ts j).interior) ∧
      ⋃ i, (Ts i).closedInterior = T.closedInterior

A triangle isn't cuttable into zero triangles.

@[category API, AMS 5 51]
lemma IsCuttable.ne_zero (hT : IsCuttable n T) : n ≠ 0 := byn:ℕT:Triangle ℝ ℝ²hT:IsCuttable n T⊢ n ≠ 0
  rintro rflT:Triangle ℝ ℝ²hT:IsCuttable 0 T⊢ False
  obtain ⟨Ts, -, -, hT⟩ := hTT:Triangle ℝ ℝ²Ts:Fin 0 → Triangle ℝ ℝ²hT:⋃ i, Simplex.closedInterior (Ts i) = Simplex.closedInterior T⊢ False
  exact T.closedInterior_nonempty.ne_empty <| byT:Triangle ℝ ℝ²Ts:Fin 0 → Triangle ℝ ℝ²hT:⋃ i, Simplex.closedInterior (Ts i) = Simplex.closedInterior T⊢ Simplex.closedInterior T = ∅ simpa using hT.symmAll goals completed! 🐙

Every triangle is cuttable into any non-zero square number of congruent triangles.

@[category API, AMS 5 51]
protected lemma IsCuttable.sq (hn : n ≠ 0) : IsCuttable (n ^ 2) T := sorry

Every triangle is cuttable into any non-zero square number of congruent triangles.


@[category API, AMS 5 51]
lemma IsCuttable.of_isSquare (hn₀ : n ≠ 0) (hn : IsSquare n) : IsCuttable n T := byn:ℕT:Triangle ℝ ℝ²hn₀:n ≠ 0hn:IsSquare n⊢ IsCuttable n T
  obtain ⟨n, rfl⟩ := hnT:Triangle ℝ ℝ²n:ℕhn₀:n * n ≠ 0⊢ IsCuttable (n * n) T; rw [← sqT:Triangle ℝ ℝ²n:ℕhn₀:n * n ≠ 0⊢ IsCuttable (n ^ 2) TT:Triangle ℝ ℝ²n:ℕhn₀:n * n ≠ 0⊢ IsCuttable (n ^ 2) T]T:Triangle ℝ ℝ²n:ℕhn₀:n * n ≠ 0⊢ IsCuttable (n ^ 2) T; exact .sq <| byT:Triangle ℝ ℝ²n:ℕhn₀:n * n ≠ 0⊢ n ≠ 0 simpa using hn₀All goals completed! 🐙

A triangle whose side lengths and angles are integrally independent is cuttable only into a non-zero square number of congruent triangles. This is proved in [So09c].

@[category research solved, AMS 5 51]
lemma isCuttable_iff_isSquare_of_linearIndependent
    (hTsides : LinearIndependent ℤ
      ![dist (T.points 0) (T.points 1),
        dist (T.points 1) (T.points 2),
        dist (T.points 2) (T.points 0)])
    (hTangles : LinearIndependent ℤ
      ![∠ (T.points 0) (T.points 1) (T.points 2),
        ∠ (T.points 1) (T.points 2) (T.points 0),
        ∠ (T.points 2) (T.points 0) (T.points 1)]) :
    IsCuttable n T ↔ n ≠ 0 ∧ IsSquare n := byn:ℕT:Triangle ℝ ℝ²hTsides:LinearIndependent ℤ ![dist (T.points 0) (T.points 1), dist (T.points 1) (T.points 2), dist (T.points 2) (T.points 0)]hTangles:LinearIndependent ℤ
  ![∠ (T.points 0) (T.points 1) (T.points 2), ∠ (T.points 1) (T.points 2) (T.points 0),
    ∠ (T.points 2) (T.points 0) (T.points 1)]⊢ IsCuttable n T ↔ n ≠ 0 ∧ IsSquare n
  exact ⟨fun hT ↦ ⟨hT.ne_zero, sorry⟩, fun hn ↦ .of_isSquare hn.1 hn.2⟩All goals completed! 🐙

Which triangles can only be decomposed into a square number of congruent triangles?

@[category research open, AMS 5 51]
lemma erdos_633 : T ∈ (answer(sorry) : Set <| Triangle ℝ ℝ²) ↔
    ∀ n, IsCuttable n T → IsSquare n := sorry

variable (n T) in

A triangle is n-simili-cuttable if it can be decomposed into n similar triangles.

def IsSimiliCuttable (n : ℕ) (T : Triangle ℝ ℝ²) : Prop :=
  ∃ Ts : Fin n → Triangle ℝ ℝ²,
    (∀ i j, (Ts i).points ∼ (Ts j).points) ∧
      Pairwise (fun i j ↦ Disjoint (Ts i).interior (Ts j).interior) ∧
      ⋃ i, (Ts i).closedInterior = T.closedInterior

A triangle isn't simili-cuttable into zero triangles.

@[category API, AMS 5 51]
lemma IsSimiliCuttable.ne_zero (hT : IsSimiliCuttable n T) : n ≠ 0 := byn:ℕT:Triangle ℝ ℝ²hT:IsSimiliCuttable n T⊢ n ≠ 0
  rintro rflT:Triangle ℝ ℝ²hT:IsSimiliCuttable 0 T⊢ False
  obtain ⟨Ts, -, -, hT⟩ := hTT:Triangle ℝ ℝ²Ts:Fin 0 → Triangle ℝ ℝ²hT:⋃ i, Simplex.closedInterior (Ts i) = Simplex.closedInterior T⊢ False
  exact T.closedInterior_nonempty.ne_empty <| byT:Triangle ℝ ℝ²Ts:Fin 0 → Triangle ℝ ℝ²hT:⋃ i, Simplex.closedInterior (Ts i) = Simplex.closedInterior T⊢ Simplex.closedInterior T = ∅ simpa using hT.symmAll goals completed! 🐙

Every triangle is simili-cuttable into any number of similar triangles, except 0, 2, 3, 5. This is proved in [So09].

@[category research solved, AMS 5 51]
lemma IsSimiliCuttable.of_ne_zero_two_three_five (hn₀ : n ≠ 0) (hn₂ : n ≠ 2) (hn₃ : n ≠ 3)
    (hn₅ : n ≠ 5) : IsSimiliCuttable n T := sorry

There exists a triangle which isn't simili-cuttable into 0, 2, 3, 5 parts. This is proved in [So09].

@[category research solved, AMS 5 51]
lemma exists_isSimiliCuttable_iff_ne_zero_two_three_five :
    ∃ T, ∀ n, IsSimiliCuttable n T ↔ n ≠ 0 ∧ n ≠ 2 ∧ n ≠ 3 ∧ n ≠ 5 := sorry

end Erdos633