Basic geometric proof of the relation between dimensionality of a regular simplex and its dihedral angle Raffaele Salvia Abstract The formula for the dihedral angle of the simplex of n dimensions, cos-1(1/n), is derived using classical geometry.
1. Introduction Several proofs are available for the theorem which states that the angle between two intersecting 1 hyperfaces of an n-simplex, namely its dihedral angle, is cos-1 . Krasnodebski (1971 [1]) first n proved the general case by using a loxodromic sequence of tangent spheres. Parks and Wills (2002 [2]) drafted a simple proof based on linear algebra. The aim of this article is to present an elementary, purely geometric proof of this result.
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2. Hypoteses As well as the common theorems of plane geometry, here are taken as given the following statements: Proposition 1. For every nonnegative number of dimensions n, there exist a regular nsimplex. Proposition 2. All the altitudes of a regular simplex are congruent and meet in one point, internal to the simplex. This point is called the orthocenter of the simplex. If n=1 (and the simplex is a segment), the midpoint is here considered as orthocenter. Proposition 3. The feet of the altitudes of a regular n-simplex coincide with the orthocenters of its (n-1)-dimensional bases.
3. Proof The proof here given is essentially based on the coincidence between orthocenter and centroid of a regular simplex. Definition 4. Call A a vertex of a simplex, H the foot of the altitude relative to A, and O the simplex's orthocenter. The regular simplex of n-dimensions is called well-built if and only if AO =n⋅OH . Proposition 5. If the n-dimensional regular simplex is well-built, also the (n+1)-dimensional regular simplex is well-built. Proof: 5.1 Be αn the n-simplex we assume being well-built. 5.2 Construct the regular (n+1)-simplex αn+1, with αn as base. 5.3 Consider the plane π identified by one of the altitudes of αn and by the altitude of αn+1 relative to the base αn . Call these two altitudes respectively AB and CF, with A and C vertexs and B and F feet (figure 1). 5.4 By proposition 2, follows that the orthocenter of αn+1 - here labeled as D - is in CF, which is in π; therefore also the altitude of αn+1 relative to the vertex A belongs to π. Call this altitude 1
AE. 5.5 Since αn+1 is regular, the hyperface α'n , containing points B and C, is congruent to αn5.6 By proposition 3 follows that E is the orthocenter of α'n. 5.7 Since 5.6, CB is an altitude of α'n.
Figure 1. 5.8 Extend CB of a segment BG such that BG≅ EB. 5.9 Draw segments BD, GF and EF (figure 2). 5.10 The intersection point of BD and EF is named H.
Figure 2. 2
5.11 AB≅CB, because altitudes of congruent n-simplexs. n n AB and CE= CB 5.12 By hypotesis, AF = n+1 n+1 5.13 Since 5.11 and 5.12, AF≅CE. 5.14 EB≅BF, because they are differences of congruent segments. 5.15 BG≅BF, since 5.8 and 5.14 (transitivity of congruence). 5.15 AE≅CF, because they are both altitudes of αn+1. 5.16 For the same reason of 5.15, ∠CFA≅∠CFB≅∠AEC≅∠AEB=90°. 5.17 ∠ADF≅∠CDE, because they are vertically opposite angles. 5.18 ΔADF≅ ΔCDE, since 5.13, 5.16 and 5.17 (leg-angle congruence of right triangles). 5.19 DF≅DE, whereas they are corresponding sides of congruent triangles . 5.20 Draw a circumference c with center B and ray BG (figure 3).
Figure 3. 5.21 Since 5.8 and 5.15, points F and E belong to c, and EG is a diameter of the circumference. 5.22 ∠ADF≅90°, because of Thales' theorem. 5.23 ΔADF≅ ΔCDE, because of 5.14 and 5.19 (side-side-side congruence of a triangle). 5.24 FH≅HE, being corresponding altitudes of congruent triangles. 5.25 Triangles ΔFBE and ΔFBG are isosceles, respectively since 5.14 and 5.15. 5.26 ∠EHB=90°, because of 5.24 (the base-relative median of an isosceles triangle is also an altitude). 5.27 ∠BGF≅∠BFG, being base angles of the isosceles triangle ΔFBG (5.25). 5.28 ∠BFE=∠EFG-∠BFG=90°-∠BFG (5.22). 5.29 ∠EFB≅∠FEB, because they are, since 5.25, base angles of an isosceles triangle. 5.30 By 5.26 follows that ΔEHB is a right triangle, and then that ∠EBH is complementary to ∠HEB. 5.31 ∠EBH≅90°-∠HEB≅90°-∠EFB≅90°-(90°-∠BFG)≅90°-90°+∠BFG≅∠BFG (lemmas from 5.27 to 5.30). 5.32 Since 5.31, lines DB and FG form, with the transversal CG, congruent correspondent 3
angles. Therefore, DB∥FG. 5.33 The intercept theorem and 5.32 lead to conclude that CD : DF =CB : BG . 5.34 Replacing in agreement with 5.8 and the well-builtness of α'n, CD : DF =CB : EB=(CB+EB): EB=( n⋅EB+EB ): EB=(n+1)⋅EB : EB . 5.35 Dividing by EB , CD : DF =(n+1):1 . So αn+1 is well-built. ∎ Proposition 6. With n≥2, every regular simplex is well-built. Proof: 6.1. In the regular 2-simplex, that is the regular triangle, medians coincide with altitudes and orthocenter coincides with centroid. 6.2. The centroid of a triangle divides every median in two parts, of which the one with the vertex is twice the other one [3]. 6.3. Since 6.1 and 6.2, the regular 2-dimensional simplex is well-built. 6.4. From lemma 6.3, preposition 5 and the principle of induction, follows that every nsimplex with n≥2 is well-built. ∎ Theorem 7. The dihedral angle of a regular n-dimensional simplex is cos-1
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Proof: 7.1. Consider any regular simplex αn (to talk about angles, the number of dimensions must be at least 2). 7.2. Since proposition 6, αn is well-built, as well as its (n-1)-dimensional faces. 7.3. In reference to figure 2, the angle ∠CBF, as angle between two hyperfaces of αn, is a dihedral angle of αn. 7.4. ΔCFB is a right triangle, because of 5.16. 7.5. The ratio between the side of ΔCFB adjacent to ∠CBF and the hypotenuse of ΔCFB, namely the cosine of ∠CBF, is FB :CB=EB :CB=CD : DF=1 :n (5.14; 5.33; 5.35, opportunely replacing n+1 with n). 1 7.6. As consequence of 7.5, ∠CBF is the inverse cosine of .∎ n This is what was wanted to prove. Corollary 8. In a regular n-simplex, the angle between two segments jointings the orthocenter 1 with two vertices is cos-1 − . n Proof: 8.1. In the construction described in 5.3 and 5.4, the angle ∠CDA separates segments AD and AC, which connect the orthocenter D with vertices A and C. 8.2. AF is an altitude of ΔCDA, for 5.16. It falls outside the base CD. 1 8.3. Because of preposition 6, the ratio between DF and CD is . n DF = -1 1 . 8.4. Since 8.2 and 8.3, the amplitude of ∠CDA is cos-1 − cos ∎ DC n
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4. References [1] Krasnodebski, W. (1971). "Dihedral angle of the regular n-simplex". Roczniki Polskiego Towarzystwa Mateatycznego, Seria I – Commentationes Mathematicae, Prace Matematyczne 15: pp. 87-89 [2] Parks, Harold R.; Wills, Dean C. (2002). "An Elementary Calculation of the Dihedral Angle of the Regular n-Simplex". The American Mathematical Monthly, 109(8): pp. 756-758 [3] Casey, J. (1888). "A sequel to the first six books of the Elements of Euclid, containing an easy introduction to modern geometry, with numerous examples." Dublin: Hodges, Figgis, & Co.: p. 3 4
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