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