Euclidean Figures and Solids without Incircles or Inspheres

Transcription

1 Foum Geometicoum Volume 16 (2016) FOUM GEOM ISSN Euclidean Figues and Solids without Incicles o Insphees Dimitis M. Chistodoulou bstact. ll classical convex plana Euclidean figues that possess incicles have aeas = p/2, whee p is the peimete, is the adius of the incicle, and the facto of 2 epesents the dimension of the space. Similaly, all classical convex Euclidean solids that possess insphees have volumes V = S/3, whee S is the total suface aea, is the adius of the insphee, and the facto of 3 epesents the dimension of the space. Elementay figues such as paallelogams and tapezoids without an incicle still obey the same aea elation, but then is the hamonic mean of the adii of the two intenally tangent cicles to opposite sides. Similaly, common solids without an insphee (notably cylindes and pisms) still obey the same volume elation, but then is the hamonic mean of the thee intenally tangent sphees to thei faces. 1. Intoduction Coxete and Geitze [2] show ealy in thei book that the aea of a tiangle can be expessed as = s, whee s is the semipeimete and is the inadius. This esult can also be witten as [1] = 1 p, (1) 2 whee p =2sis the peimete of the tiangle. The use of s by [2] implies that the authos did not intend to compae thei esult to othe classical esults obtained fo egula polygons (e.g., = pa/2, whee a is the apothem), pobably because the equation = s is valid fo any tiangle, egula o not. On the othe hand, some compaisons have been made in pint between objects and figues that ae egula, such as egula polygons and egula polyheda (e.g., owland 2015), whee it was noticed that the equations = pa/2 and V = 1 S, (2) 3 whee V is the volume and S is the total suface aea, ae pecisely simila except fo the factos of 2 and 3 that appea because of the dimension of the space in each case. This investigation stated with the following question: Which global popety does the atio /p o V/S epesent in any Euclidean figue o solid, espectively? Publication Date: July 22, Communicating Edito: Paul Yiu.

2 292 D. M. Chistodoulou b c h c a Figue 1. Isosceles tapezoid with bases a, b, legs c, inadius, and altitude h =2. These atios epesent a popety with dimensions of length that should be chaacteistic of the entie object and not of any paticula side and altitude. The above classical esults indicate that fo figues with an inscibed cicle, the atio /p is elated to the inadius; similaly, fo thee-dimensional solids with an insphee, the atio V/S is elated to the inadius of the insphee. This includes any hombus, squae, egula polygon, and one special isosceles tapezoid in two dimensions; and any cube, cone, and egula pyamid in thee dimensions, espectively. Notably absent fom the above lists ae paallelogams, ectangles, and abitay tapezoids; as well as ectangula pisms and ight cylindes in thee dimensions. So we set out to investigate such common cases in Euclidean geomety in which an incicle o an insphee cannot be dawn. Fo these cases, the above equations (1) and (2) ae still valid, except that the inadius has to be eplaced by the hamonic mean of the adii of intenally tangent cicles o sphees. In 2, we analyze the atio /p in two-dimensional Euclidean figues and in 3, we analyze the atio V/S in thee-dimensional Euclidean solids. Finally, in 4, we discuss biefly and intepet ou esults in tems of the mean cuvatue [3, 5] defined by such tangent cicles and sphees fo these objects. 2. Two-Dimensional Figues Eq. (1) can be deived quite easily fo two-dimensional figues such as squaes, hombuses, and cicles. But it is not obvious how it applies to tapezoids. Symmety tells us that thee exists a special isosceles tapezoid with bases a, b, conguent sides c, and altitude h between the bases that possesses an incicle (Fig. 1). Then: Theoem 1. If eq. (1) holds fo an isosceles tapezoid, then c =(a + b)/2.

3 Euclidean figues and solids without incicles o insphees 293 h 2 =2 2 b 2 2 h 1 =2 a Figue 2. Paallelogam with sides a, b, altitudes h 1 =2, h 2 =2 2, and without an incicle. Poof. n inscibed cicle with adius must be tangent to the bases, so = h/2 (Fig. 1). Substituting this equation and the peimete p = a + b +2c into eq. (1): = 1 2 p = 1 2 (a + b +2c)h 2 = 1 ( ) a + b + c h. 2 2 (3) Compaing this esult to the well-known fomula =(a + b)h/2, we find that a+b 2 + c = a + b which implies that c =(a + b)/2. It is also inteesting to note that fo this special isosceles tapezoid, the altitude is the geometic mean of the bases, i.e., h = ab. Next we pove the convese of Theoem 1: Theoem 2. If c =(a + b)/2, then eq. (1) holds fo this isosceles tapezoid. Poof. We begin with the well-known fomula =(a + b)h/2. n inscibed cicle with adius must be tangent to the bases, so h =2 (Fig. 1). Then = 1 2 (a + b)h = 1 (a + b)2. (4) 2 Then, we ewite (a + b)2 = (a + b) +(a + b) = (a + b) +2c = p, using the hypothesis. Theefoe, (a + b)2 = p and by substitution to eq. (4), we find eq. (1). ectangles and paallelogams do not possess an incicle. Nevetheless, we can daw two diffeent cicles that will be intenally tangent between opposite paallel sides. Let and 2 be the adii of these cicles (Fig. 2). Then:

4 294 D. M. Chistodoulou P b 2 c M 2 O N d h=2 a Figue 3. Tapezoid with bases a, b, legsc, d, altitude h =2, and without an incicle. Theoem 3. In ectangles and paallelogams with two intenally tangent cicles to opposite sides, eq. (1) holds, whee is the hamonic mean of and 2. Poof. Fo ectangles of dimensions L and W, the poof is tivial given that = L/2, 2 = W/2, and p =4( + 2 ): p = LW 2(L + W ) = 2 = 1 ( ) (5) Fo paallelogams of dimensions a and b and altitudes h 1 and h 2, we find that h 1 =2 and h 2 =2 2 (Fig. 2). We daw the diagonals to patition the figue into fou tiangles of equal aea, and we sum up thei aeas to obtain the total aea: =2 1 2 a b 2 = a + b 2. (6) Since p =2(a + b), we can wite p = a + b 2 2(a + b). (7) But a = b 2 (= /2), in which case b/a = / 2. Substituting into eq. (7), we find afte some algeba that p = 1 ( ) (8)

5 Euclidean figues and solids without incicles o insphees 295 The same esult (eq. [8]) can be poven fo the abitay tapezoid without an incicle. Fist, we need to find the common cente of the two cicles that ae intenally tangent to the bases and to the legs, espectively. Let P be the point at which the legs meet when they ae extended (Fig. 3). This cente O is located at the intesection of the midsegment MN (so that it is equidistant fom the bases) and the bisecto of P (so that it is equidistant fom the legs). By constuction, O is inteio to the tapezoid. Let be the distance of O fom the bases and 2 be the distance of O fom the legs. Then: Theoem 4. In tapezoids with two intenally tangent cicles to opposite sides, eq. (1) holds, whee is the hamonic mean of and 2. Poof. Fo a tapezoid with bases a and b,legsc and d, and altitude h, we find that h =2 (Fig. 3). We daw segments fom O to the vetices of the tapezoid to patition the figue into fou tiangles, and we sum up thei aeas to obtain the total aea: = 1 2 (a + b) (c + d) 2. (9) The aea of the tapezoid can also be expessed as = 1 2 (a + b)h = 1 2 (a + b)2 =(a + b). (10) Equating the two expessions fo, we find that c + d =(a + b) / 2. Using this equation to eliminate c + d fom the peimete, we find that p = a + b + c + d = (a + b)( + 2 )/ 2. Then using this equation fo p and eq. (10), we find that p = (a + b) = 2, (11) (a + b)( + 2 )/ which is equivalent to eq. (1) with =2 2 /( + 2 ). 3. Thee-Dimensional Solids Eq. (2) can be deived quite easily fo thee-dimensional solids such as cubes and othe egula polyheda [4], and sphees. But it is not obvious how it applies to cones, cylindes, egula polygonal pyamids, and ectangula pisms. We give below only the poofs fo cones and cylindes; the othe poofs ae simila. Conside a cone of base adius, height H, and slant height L (Fig. 4). sphee of adius can always be inscibed into this cone. Then: Theoem 5. Eq. (2) holds fo any cone. Poof. Cetain adii of the sphee ae nomal to the base of the cone and to its lateal suface (Fig. 4). vetical coss-section though the vetex V and the cente of the base C contains two simila tiangles (VDOand VCB) fom which we deive the popotion = L x, (12)

6 296 D. M. Chistodoulou V V L x L D H D O O C B C B Figue 4. Cone of adius, height H, and slant height L with an insphee of adius. Tiangles VDOand VCBon the coss-section VBae simila ( Similaity Postulate). whee x is the hypotenuse of the smalle tiangle (see Fig. 4). This is used in the total suface aea S of the cone to eliminate ( L: S = π 2 + πl = π 2 1+ x ) = π2 H, (13) since + x = H. This equation implies that π 2 H = S. Using this equation into the volume fomula of the cone V = π 2 H/3, then eq. (2) is obtained. 2 = = H=2 2 2 Figue 5. Cylinde of adius =, height H =2 2, and without an insphee.

7 Euclidean figues and solids without incicles o insphees 297 Thee also exists one special cylinde in which a sphee of adius can be inscibed. Its height then is H =2. Eq. (2) holds fo that cylinde too, but the poof is tivial and is omitted. In geneal, a sphee cannot be inscibed in an abitay cylinde. In that case, we conside a cylinde with base adius and height H in which we can daw two sphees: one intenally tangent to the lateal suface with adius = ; and anothe intenally tangent to the bases with adius 2 = H/2 (Fig. 5). Then: Theoem 6. In cylindes with two such intenally tangent sphees, eq. (2) holds, whee is the hamonic mean of,, and 2. Poof. We use 2 = H/2 to eliminate the height fom the well-known fomulae fo V and S fo a cylinde. Then we obtain V =2π1 2 2, S =2π ( +2 2 ), and thei atio is ( V S = 2 = ). (14) adius entes twice in the hamonic mean because the sphee that is intenally tangent to the lateal suface coves two of the thee pincipal diections in theedimensional space. This is undestood when eq. (14) is compaed to the coesponding esult fo an abitay ectangula pism that has thee diffeent intenally tangent sphees with adii, 2, and 3 : 4. Discussion V S = 1 3 ( ) (15) 3 We have shown that eqs. (1) and (2) hold fo a vaiety of classical Euclidean figues and solids, espectively. In cases whee an incicle o an insphee cannot be dawn, the adius in these equations evets to a hamonic mean of the adii of the cicles o sphees that ae intenally tangent to the sides o faces of the figues o solids, espectively. We note that no such geneal esults can be obtained by using the adii of the cicumscibed cicles and sphees. Theefoe, it appeas that the inadius is the most impotant global popety of these objects in any dimension; and that the atio of content to bounday (/p o V/S) is equal to /N, whee N 2 is the dimension of space. The concept of mean cuvatue K m 1 N N K i, (16) fom diffeential geomety [3, 5] allows us to ewite the equations in all cases as p = = p, (17) 2K 2 m i=1

8 298 D. M. Chistodoulou and as S V = = S, (18) 3K 3 m whee the ecipocals of the adii epesent the pincipal cuvatues K i in the N dimensional space. We see now that the atios /p and V/S (of content to bounday) ae equal to Content Bounday = ρ m N, (19) whee ρ m 1, (20) K m is the adius of the mean cuvatue K m in each object. This answes the question posed in the Intoduction. Eq. (17) is only appoximately valid fo an ellipse with semi-axes a = and b = 2. Similaly, in the case of a cicula secto with an incicle, eq. (1) is appoximate and becomes moe accuate in the limit of small opening angles θ (i.e., fo θ 0). But the eos ae emakably small in both of these cases. We believe that the poblem with the ellipse is that the cicle about the two vetices is nowhee inteio to the figue. nd the poblem with the cicula secto is that its peimete combines two linea segments with an ac. Howeve, as θ 0, the ac length appoaches that of a staight line segment and this is why the eo tem then becomes minimal. efeences [1]. G. Bown, dvanced Mathematics, 2003, McDougal-Littell, Evanston IL. [2] H. S. M. Coxete and S. L. Geitze, Geomety evisited, 1967, Math. ssoc. meica, Washington, DC. [3]. Gay, Nomal Cuvatue, in Moden Diffeential Geomety of Cuves and Sufaces with Mathematica, 2nd ed., 1967, CC Pess, Boca aton, FL. [4] E. owland, owland/investigations/ polyheda-poject.html, [5]. Schneide, Convex Bodies: The Bunn-Minkowski Theoy, 2nd ed., 2014, Cambidge Univ. Pess, Cambidge, UK. Dimitis M. Chistodoulou: Depatment of Mathematical Sciences, Univesity of Massachusetts Lowell, Lowell, Massachusetts 01854, US addess: dimitis chistodoulou@uml.edu