6 On the nuber of regons n an -densonal space cut by n hyperplanes Chungwu Ho and Seth Zeran Abstract In ths note we provde a unfor approach for the nuber of bounded regons cut by n hyperplanes n general poston n R and the nuber of regons cut by n great crcles n general poston n S Key words: Hyperplanes n R, n-cluster, n general poston, bnoal coeffcents MSC: Prary: 5C45, Secondary: 5C35, 5D Introducton In ths note we wll use the ter hyperplane n R to ean an affne subspace of codenson one n R, e, an (-densonal plane n R whch does not necessarly pass through the orgn It s well known that n hyperplanes, n general poston n the -densonal space R, wll dvde R nto regons, where ( p q f q > p In ths note, we wll nvestgate two related probles: ( What s the nature of these regons? ( How any regons occur f these hyperplanes, n general poston, happen to contan a coon pont? For the frst proble, we wll show that aong the regons cut by n hyperplanes n general poston n R, exactly ( are bounded, and we wll descrbe the geoetrcal nature of the unbounded regons For the second proble we ay assue wthout loss of generalty that the coon pont of these n hyperplanes s the orgn of R, and the proble s equvalent to that of fndng the nuber of regons on an ( -densonal sphere cut by n great crcles We wll prove that there are [( ( ( ( ] such regons We would lke to pont out that our an results could be derved fro the classcal paper of J Stener [4] However, n our paper, we wll gve a ore drect proof, and our ethod brngs out the geoetrcal relatonshps aong the followng three tes: ( the regons cut by n hyperplanes n R, all of whch contan a partcular pont; ( the bounded regons cut by n hyperplanes n general poston n R ; and ( all the regons cut The authors would lke to thank the referee for suggestons splfyng the proof of theore and enhancng the presentaton
On the nuber of regons n an -densonal space cut by n hyperplanes 6 by hyperplanes n general poston n R (see Theore and Step A n the proof of Theore below Prelnares Let us frst fx our notatons A subset A of R wll be called a k-densonal affne subset of R f for soe pont a A, the set A a ( {x a x A} s a k-densonal subspace of the lnear space R Thus, a k-densonal affne subset of R s a translate of a k-densonal subspace of R In partcular, a hyperplane n R s an affne set of codenson one, e, an -densonal plane n R whch does not necessarly pass through the orgn A set S of hyperplanes n R s sad to be n general poston f for each k, k, no k ebers of S contan a coon ( k-densonal affne subset of R If a set S of n hyperplanes n R n general poston all contan a coon pont, we wll call such a set an n-cluster Fnally, we wll let G (n be the nuber of regons n R dvded by n hyperplanes n general poston and C (n be the nuber of regons dvded by an n-cluster n R If S s an densonal sphere centered at the coon pont of an n-cluster, each eber of the cluster wll cut S n a great crcle Thus, the nuber C (n wll equal to the nuber of regons on an ( sphere S cut by n great crcles on S The followng result s well-known (see for nstance, [4] and [5], also cf [],[], and [3] Proposton G (n 3 Man results Our an results ay now be suarzed n the followng two theores Theore The nuber of regons C (n, cut by an n-cluster n R (or cut by n great crcles n S, s exactly twce G (, that s [( ( ( ( ] C (n In the followng, we wll call a regon D n R cone-lke f whenever an (-densonal plane H cuts D nto two coponents, D and D, such that the cross-secton H D s bounded, one of the two coponents D and D wll be bounded and the other unbounded For nstance, an nfnte tube, open at both ends, would not be cone-lke Theore For any set of n planes n general poston n R, aong the G (n regons cut by these n planes, exactly ( of the are bounded, and each of the unbounded regons n G (n s always cone-lke Snce the nuber of bounded regons n an -densonal space R s (, for ths nuber to be postve, Hence, there has to be at least hyperplanes n R before there can be any bounded regon at all Intutvely, therefore, there sees to ore unbounded regons than bounded ones when the space R s cut by n hyperplanes However, we wll prove that as the nuber n of hyperplanes ncreases, the nuber of bounded regons ncreases uch faster than that of the unbounded regons Specfcally, we have the followng Corollary Consder a set of n planes n general poston n an -densonal space R As the nuber n of the hyperplanes ncreases, the rato of the nuber of bounded regons over that of the unbounded regons approaches to nfnty
6 Chungwu Ho and Seth Zeran 4 Proofs of the an results Proof of Theore Snce the nuber of regons C (n, cut by an n-cluster n R, s the sae as that cut by n great crcles n S, we wll consder n great crcles n general postons n S Wthout loss of generalty, we ay assue that one of these great crcles s the equator of S Then the radal projecton wll be a bjecton between the reanng great crcles and the hyperplanes n general postons n the two tangent planes to the two poles of the sphere Ths bjecton wll also carry the regons between the great crcles to those between the hyperplanes Snce there are two tangent planes, C (n, the nuber of regons cut by n great crcles n S, equals G (, or by Proposton, [ C (n ] Proof of Theore Consder an arbtrary set C of n planes n general poston n R We frst prove that aong the G (n regons cut by these n planes, exactly ( of the are bounded Let us ebed R as an -densonal plane H n the space R so that H s one unt away fro the orgn o of R For each ( -densonal plane P n H that s a eber of C, we let P be the -densonal plane n the space R that contans both o and P The collecton C of all such P s s an n-cluster n R snce they all contan o Furtherore, they cut H n ebers of C We wll now coplete ths part of our proof n two steps, A and B Step A: Aong the G (n regons on H cut by the n planes n C exactly G (n C (n of the are bounded We can see ths as follows Let v be the unt noral vector that extends fro o to the plane H Now, let H be the plane H v, and for each t, t, let H t be the plane H tv Thus, each H t s a parallel translate of the plane H, H H, and H contans the orgn o Now, for each t, the planes P s n the collecton C ntersect H t n a collecton of -densonal planes n general poston, and thereby cut H t nto G (n regons As t changes, the boundares of these regons change contnually wth t Now, when t, soe of these regons ay shrnk to a pont At t, the n planes n the collecton C ntersect H n an n-cluster, whch cuts H nto C (n regons Thus, when t oves fro a postve value to zero, exactly G (n C (n regons are elnated Snce the regons ust change contnuously and all the regons cut by an n-cluster are unbounded, only the bounded regons are elnated Thus, aong the G (n regons on H cut by the n planes n C, exactly G (n C (n of the are bounded Step B: G (n C (n Fro Proposton and Theore G (n C (n ( n
On the nuber of regons n an -densonal space cut by n hyperplanes 63 and Note that [ ] j j, by Pascal s trangle By substtutng these two n the precedng lne, we have G (n C (n (, and aong the G (n regons, exactly ( of the are bounded To see that each of the unbounded regons s always cone-lke, consder an arbtrary set C of n planes n general poston n R Let R be the collecton of regons n R cut by these n planes Suppose D s an unbounded regon n R Let H be an ( -densonal plane that cuts D nto two coponents, D and D, such that the cross-secton H D s bounded Snce D s tself unbounded, to show that D s cone-lke, t s suffcent to show that one of D and D s bounded Tltng H slghtly f necessary, we ay obtan an (-densonal plane H whch cuts D nto two coponents, D and D, and whch s n general poston wth respect to the planes n C By pushng H slghtly to one sde or the other of H, we can ake D D or D D Thus, f we can show that one of D and D s bounded, then one of D and D wll also be bounded In the followng we ay hence assue that the plane H s n general poston wth respect to the planes n C and consder D and D nstead of D and D Now, C H {P H P C } s a set of n planes n the ( space H that cuts H nto G (n regons, each of whch dvdes a regon of R nto two Thus, wth the addton of H, G (n new regons have been added Now, each of the unbounded regons aong the G (n regons on H can add only a new unbounded regon n R, not a bounded one Thus, the new bounded regons n R can coe only fro bounded regons of H Before the addton of H, there were already ( bounded regons n R cut by the ebers of C, and after H s added, there are ( ( n bounded regons Thus, there are n ( ( new regons beng added But there are only ( bounded regons on H Each of these bounded regons ust therefore create a new bounded regon n R But the cross-secton H D s one such bounded regon whch cuts an old regon D nto two coponents, D and D Thus, one of D and D ust be bounded Proof of Corollary Consder a set of n planes n general poston n an -densonal space R For a fxed, the nuber of bounded regons, (, has an order of agntude O(n On the other
64 hand, the nuber of unbounded regon, fro A n the proof of Theore, s C (n, but [( ( ( ( ] C (n It has an order of agntude of O ( O(n Thus, the order of agntude of the bounded regons s greater than that of the unbounded regons and the rato of these two nuber approaches nfnty as n ncreases References [] G L Alexanderson and J E Wetzel, Sple dvsons of space, Matheatcs Magazne, 5 (978, 5 [] RC Buck, Parttonng of space, Aercan Matheatcal Monthly, 5 (943, 54 544 [3] J W Kerr and J E Wetzel, Platonc dvsons of space, Matheatcs Magazne, 5 (978, 9 34 [4] J Stener, Enge Gesetze ber de Thelung der Ebene und des Raues, J Rene Angew Math (86, 349 364 [5] S Zeran, Slcng space, The College Matheatcs Journal, 3 (, 6 8 Evergreen Valley College, San José, CA 9535 and Southern Illnos Unversty, Edwardsvlle, IL 66 E-al: HoC@alutedu Evergreen Valley College, San José, CA 9535 E-al: sethzeran@sjeccdorg Receved Septeber 5, accepted for publcaton 7 Noveber 5