| import OAI.Geometry.CylinderCovering.Beams.FullDefinitions |
| import OAI.Geometry.CylinderCovering.Convexity.RegularCounterexamples |
| import OAI.Geometry.CylinderCovering.Simplex.ProjectionIdentity |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open Set MeasureTheory |
| namespace ShortenedBeam |
| open CylinderCovering |
|
|
| private lemma root_two_ne : Real.sqrt 2 ≠ 0 := by positivity |
| private lemma root_two_sq : (Real.sqrt 2)^2 = 2 := Real.sq_sqrt (by norm_num) |
|
|
| def normalizationMap : Space → Space := |
| Similarity.map (point 0 0 (-1)) (Real.sqrt 2) (LinearIsometryEquiv.refl ℝ Space) |
|
|
| lemma normalizationMap_point (x y t : ℝ) : |
| normalizationMap (TriangularCovering.point x y t) = point (2*x) (2*y) (2*t-1) := by |
| ext i |
| fin_cases i <;> |
| simp [normalizationMap, Similarity.map, point, TriangularCovering.point, |
| TriangularCovering.H] <;> field_simp <;> ring_nf <;> simp <;> ring |
|
|
| lemma tetrahedron_eq_normalization : |
| tetrahedron = normalizationMap '' TriangularCovering.K := by |
| ext z |
| constructor |
| · rintro ⟨x,y,t,ht,hx,hy,rfl⟩ |
| refine ⟨TriangularCovering.point (x/2) (y/2) ((t+1)/2), ?_, ?_⟩ |
| · refine ⟨x/2,y/2,(t+1)/2,?_,?_,?_,?_,rfl⟩ |
| · linarith [(abs_le.mp ht).1] |
| · linarith [(abs_le.mp ht).2] |
| · rw [abs_div]; norm_num; linarith |
| · rw [abs_div]; norm_num; linarith |
| · rw [normalizationMap_point] |
| congr 1 <;> ring |
| · rintro ⟨_,⟨x,y,t,ht0,ht1,hx,hy,rfl⟩,rfl⟩ |
| rw [normalizationMap_point] |
| refine ⟨2*x,2*y,2*t-1,?_,?_,?_,rfl⟩ |
| · rw [abs_le]; constructor <;> linarith |
| · rw [abs_mul]; norm_num; linarith |
| · rw [abs_mul]; norm_num; linarith |
|
|
| lemma vertices_eq_normalization : |
| vertices = normalizationMap ∘ TriangularCovering.tetraVertices := by |
| funext i |
| fin_cases i <;> |
| norm_num [vertices, TriangularCovering.tetraVertices, Function.comp_def, |
| normalizationMap_point] |
|
|
| lemma tetrahedron_eq_vertices : tetrahedron = convexHull ℝ (range vertices) := by |
| let f := Similarity.affineMap (point 0 0 (-1)) (Real.sqrt 2) |
| (LinearIsometryEquiv.refl ℝ Space) |
| rw [tetrahedron_eq_normalization, TriangularCovering.K_eq_convexHull, |
| vertices_eq_normalization] |
| change f '' convexHull ℝ (range TriangularCovering.tetraVertices) = |
| convexHull ℝ (range (f ∘ TriangularCovering.tetraVertices)) |
| rw [f.image_convexHull, range_comp] |
|
|
| lemma vertices_affineIndependent : AffineIndependent ℝ vertices := by |
| rw [vertices_eq_normalization] |
| exact TriangularCovering.tetra_affineIndependent.map' |
| (Similarity.affineMap (point 0 0 (-1)) (Real.sqrt 2) |
| (LinearIsometryEquiv.refl ℝ Space)) |
| (Similarity.map_injective _ root_two_ne _) |
| |
| lemma vertices_edge_length {i j : Fin 4} (hij : i ≠ j) : |
| dist (vertices i) (vertices j) = 2*Real.sqrt 2 := by |
| rw [vertices_eq_normalization] |
| simp only [Function.comp_apply, normalizationMap, Similarity.map, |
| dist_add_left] |
| rw [dist_smul₀, Real.norm_eq_abs, abs_of_pos (by positivity : 0 < Real.sqrt 2)] |
| change Real.sqrt 2 * dist (TriangularCovering.tetraVertices i) |
| (TriangularCovering.tetraVertices j) = _ |
| rw [TriangularCovering.tetra_edge_length hij] |
| ring |
| |
| lemma tetrahedron_minimum_shadow : |
| minimumShadowArea tetrahedron = 2*Real.sqrt 2 := by |
| change Similarity.minimumArea tetrahedron = _ |
| rw [tetrahedron_eq_normalization] |
| change Similarity.minimumArea (Similarity.map _ _ _ '' TriangularCovering.K) = _ |
| rw [Similarity.minimumArea_map _ _ root_two_ne] |
| change (Real.sqrt 2)^2 * TriangularCovering.A_min = _ |
| rw [root_two_sq, TriangularCovering.projection_minimum] |
| rfl |
| |
| lemma tetrahedron_regular : IsRegularTetrahedron tetrahedron := by |
| exact ⟨vertices, vertices_affineIndependent, 2*Real.sqrt 2, by positivity, |
| fun _ _ h => vertices_edge_length h, tetrahedron_eq_vertices⟩ |
| |
| theorem geometry_of_volume (hvol : volume.real tetrahedron = 8/3) : GeometryStatement := |
| ⟨tetrahedron_eq_vertices, vertices_affineIndependent, |
| fun _ _ h => vertices_edge_length h, hvol, tetrahedron_minimum_shadow⟩ |
| |
| lemma tetrahedron_volume : volume.real tetrahedron = 8/3 := by |
| rw [tetrahedron_eq_vertices] |
| rw [CylinderCovering.regular_tetrahedron_volume vertices vertices_affineIndependent |
| (show 0 < 2*Real.sqrt 2 by positivity) (fun _ _ h => vertices_edge_length h)] |
| have hs : Real.sqrt 2 ≠ 0 := root_two_ne |
| field_simp |
| nlinarith [root_two_sq] |
| |
| theorem geometryStatement : GeometryStatement := geometry_of_volume tetrahedron_volume |
| |
| end ShortenedBeam |
| |
| end |
| |
| end OAI |
| |
