Visualization of Gauss-Bonnet Theorem

Transcription

1 Visualizatio of Gauss-Boet Theorem Yoichi Maeda Departmet of Mathematics Tokai Uiversity Japa Abstract: The sum of exteral agles of a polygo is always costat, π. There are several elemetary proofs of this fact. I the similar way, there is a ivariat i polyhedro that is 4 π. To see this, let us cosider a regular tetrahedro as a example. Tetrahedro has four vertices. Three regular triagles gather at each vertex. Developig the tetrahedro aroud each vertex, there is a ope agle, π. The sum of these ope agles is 4 π. As aother example, let us cosider a cube. There are eight vertices ad a ope agle is π / at each vertex. The sum of ope agles is also 4 π. This fact is regarded as a discrete case of the famous Gauss-Boet theorem. Usig dyamic geometry software Cabri D, we ca easily uderstad a simple proof of this theorem. The key word is polar polygo i spherical geometry.. Itroductio Elemetary school studets kow the sum of exteral agles of polygos as well as the sum of iterior agles. The sum of exteral agles of polygos is always π. I this sese, polygo has a ivariat. How about a ivariat of polyhedro? That is the value of 4 π. This is a special case of the famous Gauss-Boet theorem, however, we ca easily show elemetary school studets this ivariat. To see this, let us defie ope agle at first. Defiitio.(Ope Agle of Polyhedro) Let V be a vertex of a polyhedro. Developig the polyhedro aroud the vertex V, there is a agle which forms π whe joied together with the agles of faces gatherig at V (Figure.). We call this complemetary agle ope agle at the vertex V. Figure. Pyramid ad ope agle at the top. Usig this ope agle, we ca easily show the ivariat i polyhedro for elemetary school studets.

2 Figure. shows a regular tetrahedro ad its et. Tetrahedro has four vertices, ad three π regular triagles gather at each vertex. The ope agle at each vertex is equal to π =. The sum of ope agles is 4 π i this case. Figure. Tetrahedro ad its polyhedral et. The ext example is a cube. Figure. shows a cube ad its et. Cube has eight vertices, ad three squares gather at each vertex. Flatteig these three faces aroud each vertex o a plae, there π π exists a ope agle = at each vertex. The sum of ope agles is also 4 π. Figure. Cube ad its polyhedral et. The ext oe more example is sufficiet for studets to asset this ivariat. Figure.4 shows a octahedro ad its et. Octahedro has six vertices, ad four regular triagles gather at each vertex. The ope agle is π = 4 π at each vertex. The sum of ope agles is 4 π, too. Figure.4 Octahedro ad its polyhedral et.

3 At this stage, studets wat to kow the reaso why the sum of ope agles is always costat. It is a little bit complicated to explai. I this paper, we will visualize this reaso with dyamic geometry software CabriD. For simplicity, we assume that polyhedros are covex ad with o geuses. We will recall the ivariat i polygo i Sectio. Two ituitive proofs are provided, ad both proofs are importat for the explaatio of the ivariat of polyhedro. I Sectio, we will see ituitively that the ivariat of polyhedro is 4 π by fatteig the polyhedros. Fially, we will see the relatio betwee the ope agles of polyhedro ad the ivariat. I this course, studets will aturally gai a ew uderstadig of difficult cocepts: spherical geometry, duality, ad curvature.. Ivariat i polygo I this sectio, we will review the ivariat i polygo; the sum of exteral agles of polygo is always π. Oe proof is showed i Figure.(left). Imagie that a vector (or a pecil) moves alog the covex polygo. At a vertex, the vector turs to the left with the exteral agle. Whe the vector returs to the start poit, the vector has rotated just oe time. Hece the sum of exteral agles is π. The exteral agle is regarded as the agle to flatte the vertex agle of the polygo o a straight lie. I this sese, exteral agle i polygo may correspod to ope agle i polyhedro. Figure. Exteral agles(left) ad Fatteig polygos(ceter ad right). Aother proof is to fatte the polygo i Figure.(ceter). Whe we fatte a polygo, there is a circular sector at each vertex ([] p.8). Here, let us prepare the followig defiitio. Defiitio.(Outer Agle of Polygo) Let V be a vertex of a polygo. Fatteig the polyhedro, there is a circular sectio at V. We call the cetral agle of the circular sector outer agle at the vertex V. Outer agle does ot deped o the size of the circular sector. Note that at each vertex, the outer agle is equal to the exteral agle. It is easy to see that outer agles form a whole disc, that is, our ivariat. The more we fatte the polygo, the more the fatteed polygo looks like circle as i Figure.(right). I this way, we ca see that circular sectors are deeply related to the ivariat. This fact is useful whe we cosider the ivariat i polyhedro i the ext sectio.

4 . Ivariat as outer solid agles I this sectio, let us cosider to fatte polyhedros, ad discover the existece of a ivariat i polyhedro. Figure. Fatteig tetrahedro. Figure. shows fatteig of tetrahedro. We ca see that there is a spheric sector at each vertex ad their uio form the ball ([] p.9). Defiitio.(Outer Solid Agle of Polyhedro) Let V be a vertex of a polyhedro. Fatteig the polyhedro, there is a spheric sectio at V. We call the solid agle of the spheric sector outer solid agle at the vertex V. Outer solid agle does ot deped o the size of the spheric sector. Outer solid agle is idetified as the correspodig area of the uit sphere. Hece the sum of these outer solid agles is 4 π. This value is our desired ivariat. Figure. also shows fatteig of cube. I the same way, eight spheric sectors at each vertex also form the ball. Figure. Fatteig cube. Our aim i the followig argumet is to show that this outer solid agle is equal to the ope agle described i Sectio. I this cotext, we ca say the relatio amog four agles symbolically; Exteral agle : Ope agle = Outer agle : Outer solid agle.

5 4. Ivariat as ope agles Oce agai, let us show two more examples. Figure 4. shows a dodecahedro ad its et. Dodecahedro has twety vertices, ad three regular petagos gather at each vertex. Flatteig π π these three faces aroud each vertex o a plae, there exists a ope agle = at 5 5 each vertex. The sum of ope agles is 4 π. Figure 4. Dodecahedro ad its polyhedral et. Figure 4. shows a icosahedro ad its et. Icosahedro has twelve vertices, ad five regular triagles gather at each vertex. Flatteig these five faces aroud each vertex o a plae, there π π exists a ope agle = 5 at each vertex. The sum of ope agles is 4 π, too. Figure 4. Icosahedro ad its polyhedral et. From Sectio, to proof the sum of ope agles of a polyhedro is always equal to 4 π, it is sufficiet to show that the ope agle is equal to the outer solid agle at each vertex. Sice solid agle is defied as the spherical area i the uit sphere, i the followig argumet, we assume that spherical sectors are i the uit ball. Figure 4. shows a vertex V with spheric sector A B C. Spherical triagle A B C is called the polar triagle of spherical triagle ABC. These two triagles are mutually dual.

6 Figure 4. Spheric sector at a vertex. I the picture above, poit B is a pole of the great circle C A(see, Figure 4.(right)). I the same way, poit C is a pole of the great circle A B, ad so o. The ote that BVC = π A. It is similar to outer agle i Sectio. Here recall the Girard s formula; the area of a spherical triagle with agles α, β, γ is α + β + γ π ([] pp.78-79, [] p.5). The, whe three faces BVC, CVA, ad AVB form the vertex V of a polyhedro, ope agle at the vertex( VABC) = ( BVC + CVA + AVB) = (( π A) + ( π B) + ( π C)) = A + B + C π More geerally, we ca show the followig propositio. = area of spehrical triagle( A B C) = outer solid agle at the vertex( VABC). Propositio 4.(Ope Agle ad Outer Solid Agle) Let V be ay vertex of covex polyhedro. The ope agle at V is equal to the outer solid agle at V. Proof. If -faces A VA, AVA,, ad A VA form the vertex V, ope agle at a vertex( VA A A L A ) ( A = + + = (( π A ) + ( π A = A + A + A VA A VA + L + A ) + ( π A ) + L + ( π A ) ( ) π = area of spehrical polygo( A A A L A ) = outer solid agle at the vertex( VA A A A VA L A + L + A ). VA ) I this way, we have proved that the sum of ope agles of covex polyhedro is 4 π.

7 5. Coclusios I this paper, we itroduce the ivariat of ope agles i polyhedro. It is ot difficult for studets to uderstad this ivariat by usig three-dimesioal geometry software. I additio, it is fouded out that the ivariat i polyhedro is a atural extesio of that i polygo. This ivariat is applicable ot oly for regular polyhedros ad also geeral polyhedros: cocave polyhedros or polyhedros with geuses. I this case, the value of ivariat is equal to χ ( M ) where M is a polyhedro ad χ(m ) is the Euler Characteristic of M (Gauss-Boet theorem). Through this study, studets will be iterested i three-dimesioal objects, spherical geometry, ad more geeral Riemaia geometry. Refereces [] Berger, M. (977). Geometry II. Spriger-Verlag Berli Heidelberg. [] Jeigs, G. (994). Moder Geometry with Applicatios. Spriger-Verlag New York.