Mathematical Constants I & II

This is a sel -a chi ed e sion o an o iginal a icle. This e sion may di e om he o iginal in pagina ion and ypog aphic de ails. Au ho (s): Ti le: Yea : Ve sion: Copy igh : Righ s: Righ s u l: Please ci e he o iginal e sion: CC BY 4.0 h ps://c ea i ecommons.o g/licenses/by/4.0/ Ma hema ical Cons an s I & II © 2019 The Au ho (s), Published e sion Pekonen, Osmo Pekonen, O. (2019). Ma hema ical Cons an s I & II. Ma hema ical In elligence , 42(1), 87-88. h ps://doi.o g/10.1007/s00283-019-09929-0 2019 Ma hema ical Cons an s, Ma hema ical Cons an s II by S e en R. Finch CAMBRIDGE UNIVERSITY PRESS, 2003 & 2019. ISBN: 9780521818056 (VOL. I), 9781108470599 (VOL. II), 624 + 782 PP., US $165.00 (VOL. 1), $175.00 (VOL. 2) REVIEWED BY OSMO PEKONEN H How many signi ican ma hema ical cons an s do you know? Mos s uden s o ma hema ics will emembe pand e, o cou se, and i p essed, he golden mean uand he Eule –Masche oni cons an c. Roge Ape ´ y made headlines in 1978 when he p o ed ha (3), he alue o he Riemann ze a unc ion a he odd posi i e in ege 3, is i a ional. An insc ip ion on his ombs one a he Pe ` e Lachaise Ceme e y in Pa is p oclaims: 1þ1 8þ1 27 þ1 64 þ6¼ p q: A gene aliza ion o Ape ´ y’s wo k o (2k+1) o k[1 s ill is ‘‘a mys e y w apped in an enigma’’ (as desc ibed in he sp ing 1979 issue o his magazine). Fo Ca alan’s cons an G¼X 1 n¼0 1ðÞ n 2nþ1ðÞ 2¼0:9159655941...; a simila i a ionali y esul is also unknown. S e en R. Finch’s inc edible labo o lo e, an encyclopedia o ma h- ema ical cons an s, begins wi h such basics, hen mo es on o mo e elabo a e opics. Some ma hema ical cons an s a e known as p ecise closed- o m exp essions, bu many a cons an mani es s i sel only as an app oxima ion known o a ew decimal places o loca ed be ween bounds whose imp o emen is a challenge. Chai in’s e asi e cons an X,o he hal ing p obabili y o a sel -delimi ing uni e sal com- pu e , exis s bu canno be compu ed. And hen he e a e pu ely conjec u al cons an s whose e y exis ence has no been es ablished. Some cons an s discussed a ise na u ally om au hen ic ma hema ical p oblems, while o he s look mo e like ec ea ions. Among he la e , le us men ion as an example T o ’s cons an E= 0.1084101512…, he unique numbe whose decimal digi s coincide wi h he sequence o coe icien s in i s expansion as a con inued ac ion. An open ques ion is whe he Eis anscenden al. One wonde s how hese wo olumes we e bo n. Pub- lished as books 94 and 169 o he Encyclopedia o Ma hema ics and I s Applica ions, hey con ain a o al o 269 me iculously documen ed essays om all ields o ma he- ma ics. To name a ew, in Volume I he e a e a icles on he s a is ics o con inued ac ions, chaos in nonlinea sys ems, p ime numbe s, sum- ee se s, isope ime ic p oblems, app oxima ion heo y, sel -a oiding walks and he Ising model ( om s a is ical physics), bina y and digi al sea ch ees ( om heo e ical compu e science), he P ouhe – Thue–Mo se sequence, uni alen unc ions, geome ic p obabili y, he a eling salesman p oblem, and bes packings. And in Volume II, one can ind essays on numbe heo y, g aph heo y, combina o ics, inequali ies and app oxima ion, ellip ic cu es and modula o ms, Poisson– Vo onoi essella ions, andom iangles, B ownian mo ion, P and l–Blasius low ( om luid dynamics), Lyapuno exponen s, kno s and angles, con inued ac ions, Gal on– Wa son ees, elec ical capaci ance ( om po en ial heo y), Ze melo’s na iga ion p oblem, and he minimali y o soap ilm su aces. I appea s as onishing o me ha a single indi idual wen h ough all hese opics. His achie emen can only be compa ed o he On-Line Encyclopedia o In ege Sequences. Could his be ye ano he case o obsessi e collec ing diso de ? By no means, o we lea n ha S e en R. Finch is also an accomplished pianis and compose , so he does ha e a li e beyond o mulas and numbe s. Some o he mos in iguing o mulas o ma hema ics (like hose o Ramanujan) ado n his easu e o e o ma hema ical gems. Open p oblems abound. Finch gen- e ously gi es c edi o obscu e pape s by bap izing some lesse -known cons an s wi h he names o hal - o go en au ho s, so his wo books se s anda ds o nomencla u e. Hono ing he elde s, he would like o call ffiffiffi 2 pPy hago as’s cons an and pA chimedes’s cons an . Well-known ma h- ema icians who ge a lo o hi s in he index o names o Volume I include he Bo wein b o he s, Conway & Sloane, E d} os, Ga dne , G adhs eyn & Ryzhik, Ha dy & Li lewood, Knu h, Plou e, Po ´lya & Szeg} o, Ribenboim, Zeilbe ge , and many o he s. Finch’s men o Philippe Flajole (1948–2011) is o en ci ed as well. Volume II un o una ely has no index o names, bu many o he same names ound in he i s olume appea he e as well. Finch s ops a he ga es o s ing heo y, men ioning some esul s in enume a i e geome y due o Kon se ich, Va a, Wi en, and o he s, bu concedes ha ‘‘much wo k lies ahead o igo ously con i m e e y hing w i en he e.’’ Amazingly, much o ma hema ics seems o be no hing bu a s o y abou p,e,u, and c. These cons an s keep appea ing unde di e en guises h oughou he wo ol- umes. Finch would like o demons a e ha o he cons an s as well migh be mo e impo an and in e connec ed han we hink. ‘‘Tha is, i we wo k and lis en ha d enough, he echoes will become audible.’’ E en so, no o he ma he- ma ical cons an s seem o gain p ominence compa able o ha o he classical qua e . Je Wimp e iewed Finch’s i s olume in his magazine (summe issue 2004), asse ing ha ‘‘all ma hema icians should own his book.’’ The same applies o he second olume. 2019 The Au ho (s), Volume 42, Numbe 1, 2020 87 h ps://doi.o g/10.1007/s00283-019-09929-0 ACKNOWLEDGMENTS Open access unding p o ided by Uni e si y o Jy a ¨skyla ¨ (JYU). OPEN ACCESS This a icle is dis ibu ed unde he e ms o he C ea i e Commons A ibu ion 4.0 In e na ional License (h p:// c ea i ecommons.o g/licenses/by/4.0/), which pe mi s un es ic ed use, dis ibu ion, and ep oduc ion in any medium, p o ided you gi e app op ia e c edi o he o i- ginal au ho (s) and he sou ce, p o ide a link o he C ea i e Commons license, and indica e i changes we e made. Facul y o In o ma ion Technology Uni e si y o Jy a ¨skyla ¨ Ago a, PL 35, 40014 Jy a ¨skyla ¨ Finland e-mail: [email p o ec ed] Publishe ’s No e Sp inge Na u e emains neu al wi h ega d o ju isdic ional claims in published maps and ins i u ional a ilia ions. 88 THE MATHEMATICAL INTELLIGENCER