Which Tetrahedra Fill Space?
Author(s): Marjorie Senechal
Source: Mathematics Magazine, Vol. 54, No. 5 (Nov., 1981), pp. 227-243
Published by: Mathematical Association of America
Stable URL: http://www.jstor.org/stable/2689983
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L1 ICLB
WhichTetrahedraFillSpace?
gave some puzzlinganswers;
Earlymathematicians
today the problemis not yet completelysolved.
MARJORIESENECHAL
SmithCollege
MA 01063
Northampton,
anygapsis one of theoldestand
without
together
polyhedra
congruent
Fillingspacebyfitting
in ancienttimes
It arosefirst
history.
and has a fascinating
problems,
of geometric
mostdifficult
ithas
2300yearsofitsdevelopment,
duringthesubsequent
ofmatter;
in relation
to Plato'stheory
of
the
in
structure
interested
and
others
fromphysicists
stimulus
to receiveitsprincipal
continued
theshapesofbuilding
is thatofdetermining
theproblem
form,
thesolidstate.In itsmostintuitive
of spaceitself.Its
and organicmatter,
of inorganic
blocks- thebuildingblocksof architecture,
thatall matteris the resultof
origincan be tracedto Plato'satomictheory:thehypothesis
questionis:
units.The mathematical
of a fewbasic polyhedral
and permutations
combinations
tothat
congruent
gapsbyfigures
whatshapemustsucha unithaveifitispossibletofillspacewithout
efforts
problemis stillunsolved,despiteconsiderable
singleunit?This simplystatedgeometric
devotedto it overtheages.
to fillspace
can be fittedtogether
parallelopipeds
solidsor,moregenerally,
Thatrectangular
is less
have thisproperty
but thatanyotherpolyhedra
was knownto theearliestbricklayers,
wasthefirst
butAristotle
ofsuchpolyhedra,
obvious.Plato,as we shallsee,assumedtheexistence
a controversy
lasting
to getdownto details.In theprocesshe made a mistakethatgenerated
nearly2,000years.
Aristotle
assertedthat,of thefiveregularsolids(FIGURE 1), notonlythecube but also the
(FIGURE 2) does not seemto havebeen evident
fillsspace.That thisis incorrect
tetrahedron
was
scholars-iftheyrealizedthatsomething
at thattime,and manyof thelaterAristotelian
Aristotle's
to
In
justify
be
mistaken.
trying
assumedthatsomehowtheymust
amiss-apparently
actuallydo fillspace,
questionofwhichtetrahedra
theyraisedtheinteresting
erroneous
assertion,
polyhedra.
usedtodayin thestudyofspace-filling
and theydevelopedsomeofthetechniques
in detailbyDirkStruikin 1925[1];
wasdiscussed
ofthespace-filling
problem
Theearlyhistory
played a
therehe showedhow Aristotle'serror,forall the confusionit caused,indirectly
in
ofpolyhedral
ofthetheory
angles.The storyis instructive
rolein thedevelopment
constructive
of a problemor excessive
misunderstanding
manyways.It showshowerrorscan arisethrough
forthesereasonsor through
simple
andhowtheycan be perpetuated
deference
to a greatthinker,
Thefirst
sectionofthis
resolved.
hasbeenproperly
evenaftertheproblem
sometimes
carelessness,
rolethatthe
sketchtheimportant
paperis based on Struik'sarticle.In thesecond,we briefly
fromabout1600
ofcrystals
of thestructure
ofthetheory
problemhas playedin thedevelopment
and
betweengeometry
to thepresent.In thisdiscussionwe hope to showthattheinteraction
forbothsides.Finally,we discussthequestioninadvertently
naturalsciencescan be profitable
do not?
whichtetrahedra
fillspaceand which
raisedbyAristotle:
VOL. 54, NO. 5, NOVEMBER 1981
227
_
~~~~~~~~CUBE
ELar/i
TETFRAHEDRON
Fbe
OCTAHEDRON/
ir*
ICOSAHEDRON
DODECAHEDRON
I al/Ier
hlieU?iivcrse
Mundi,BookII (1619). Redrawnby
solidsas depictedbyJohannesKeplerin Harmonices
FIGURE 1. The fiveregular
1 in [3]. (Reprinted
withpermission.)
JohnKyrk,thisis Illustration
?1. An errorforalmost2000 years
wasoneof
oftheregular
solids,and theproofthatthereareexactlyfiveofthem,
Thediscovery
of theancientGreeks.Theywerediscussedin detailby
thegreatmathematical
achievements
thatthepurposeofthe
ithasevenbeensuggested
Euclidin thefinalbook(XIII) oftheElements;
Platoseemsto havebeenthe
treatment
of theirconstruction.
Elements
was to providea rigorous
in theinterpretation
ofnature:theywerethebasis
of thesepolyhedra
firstto "apply"thetheory
in hisdialogueTimaeus.
to Plato,all matter
According
ofhistheory
ofmatter,
whichis presented
of fourbasic "elements":earth,air,fire,and water(thiscorresponds
consistsof combinations
ratherwell to our presentconceptof thephasesof matter).The elementsof each typeare
arecubes,
shape:theearthparticles
"fartoosmalltobe visible,"ofdefinite
composedofparticles,
theairparticles
octahedra,
andthoseoffire,regular
regular
thewaterparticles
icosahedra,
regular
was associatedwith the
tetrahedra.
(The fifthregularsolid, the pentagonaldodecahedron,
liquids)were
of theelements
kindsofstone,or different
(suchas different
cosmos.)The varieties
sizes,whilesubstances
thatthebasicparticles
comein manydifferent
on thehypothesis
explained
of the corresponding
of elementswere assumedto consistof mixtures
whichare mixtures
particles.
is composedof
withreality.
If a substance
Aristotle
is incompatible
arguedthatPlato'stheory
to fillthe space
particlesof a givenshape and size, thentheseparticlesmustpack together
FIGURE2.
does not fillspace without
The regulartetrahedron
gaps. Its fourfaces are equilateraltriangles,from
whichit followsthatits dihedralanglesa (theangles
betweenadjacentfaces)are equal to arccos(1/3), or
are fittedaroundan edge,
a 700 32'. If 5 tetrahedra
thereis a gapwhoseangularmeasure0 is less than x,
do not fill
and we concludethatregulartetrahedra
space when arrangedface-to-face.In any other
a dihedralangle of 'ir- x is created,
arrangement
whichcannotbe filledbyregulartetrahedra.
228
/
/
\
\
\
/
-
0
MATHEMATICSMAGAZINE
andregularhexagons.No otherregularpolygons
FIGURE3. The planecan be filledwithsquares,equilateraltriangles,
can filltheplanewithout
gaps.
occupiedby thesubstance,
thatis, theymustfillspacewithout
leavinganygaps.A gap would
meanemptyspace,or a vacuum,whichaccordingto theAristotelian
theoryof motioncannot
"In
occurin nature.But someof theregularsolidsdo not fillspace.Thus,remarked
Aristotle,
generalitis incorrect
to givea formto eachofthesingular
bodies,in thefirst
place,becausethey
willnotsucceedin filling
thewhole.It is agreedthatthereexistonlythreeplanefigures
thatcan
filla place, the triangle,
and the hexagon,and onlytwo solid bodies,the
the quadrilateral,
pyramidand the cube. But the theorydemandsmorethanthese,becausethe elementsthey
represent
are greater
in number"(quotedfromDe Caelo III, 306b).Thiswas considered
to be a
seriousargument
againstthe ancientatomictheory,
whichconsequently
becameincreasingly
unpopular.
We mayassumethatAristotle
wasreferring
to thefactthattheonlyregular
polygons
whichfill
are the square,the equilateraltriangle,
and the regular
the plane withcopies of themselves
hexagon(FIGURE 3). Fromthisand fromthecontextof his remark,
we concludethatAristotle
believedthattheregularoctahedron
and icosahedron
do not fillspace (in thishe was correct)
whilethecube and regulartetrahedron
do. He gaveno evidenceforhis claim.Struikremarks,
"This passage,whichis only reportedincidentally
in a modeminvestigation
of Aristotle's
mathematics,
causedtheancientwriters
considerable
concern."
Thuswe finda seriesofcommentatorson Aristotle
thenumber
oftetrahedra
thatcan "fillthespaceabouta point,"that
discussing
is,be packedtogether
so as to sharea vertex.
Simplicius,
a scholarand commentator
wholivedin
thefirst
halfofthesixthcentury
is twelve,
but
A.D., assertedthatthenumberofsuchtetrahedra
who probablylivedin the first
gave no reason.He also statedthatPotaman(a philosopher
centuryA.D.) had concludedthat the numberwas eight,by the following
reasoning.The
each of thecubes
maximum
numberof cubeswhichcan sharea vertexis eight.If we truncate
weobtaineighttetrahedra
whichfillspaceabouta point(FIGURE4).
meeting
at thatvertex,
A
E
D~~~~~~~~~~~~~
D~~~~~~
G
F
/
(a)
F
(b)
(c)
FIGURE 4. (a) Cubes can pack space, eightmeeting
at a singlevertex.(b) Each cube can be partitioned
intofive
theconstruction
Potamanhad in mind.Only
tetrahedra:
BGCD, EFBG, ABED, DEHG, and DBEG. This is probably
the"central"tetrahedron
DBEG is regular;theotherfourare congruent
"corners"of thecube. (c) Eighttetrahedra
to DEHG willpack withall rightanglesmeetingat a singlepointto forma regularoctahedron.
congruent
VOL. 54, NO. 5, NOVEMBER 1981
229
Potaman'sargument,
as reported
here,has somedefects
whichillustrate
thesortofdifficulties
its history.
In thefirstplace,theeighttetrahedra
whichhaveplaguedthisproblemthroughout
Potamanobtainsby truncating
eightpackedcubesare notregular:thetetrahedral
faceswhich
notequilateral,
WhileAristotle
did notstate
of thecubeare right,
meetat thevertices
triangles.
itis,as wehaveseen,reasonableto assumethatthisis
thathe meantregular
explicitly
tetrahedra,
whathe intended.
Evenif we ignorethisobjection,anotherremains:fillingthespace abouta
pointis notthesamethingas filling
space as a whole.Somepackingarrangements
cannotbe
continuedto fillall of space. It is possiblethatPotamanthoughtthatif his "truncation"
procedure
wascarriedouton all thecubesin a regular
packingofcubes,theneachcubewouldbe
ifthiswereso, thenthetetrahedra
intocongruent
wouldbe space-fillers.
partitioned
tetrahedra;
Whilehisconstruction
does dissecteach cubeintofivetetrahedra,
thesefivetetrahedra
are not
the"central"tetrahedron
is regular.
Also an argument
congruent:
similarto thatin thecaptionto
FIGURE2 showsthathis "vertextetrahedra"
do not fillall space. Nevertheless,
Potaman's
a knownspace-filler
technique,
intocongruent
partitioning
parts,is one of themostusefulwe
haveforconstructing
newspace-filling
polyhedra.
Aristotle's
errornot onlystimulated
of thistechnique;it also led to the
thedevelopment
earliest
attempts
to definethemeasure
ofa polyhedral
angle.The 12thcentury
Arabiccommentatoron Aristotle,
Averroes,
seekingto justifyAristotle's
remark,
developeda theoryof angle
measurewhichbeautifully
serveditspurpose.According
themeasureof a trihedral
to Averroes,
angle(suchas theangleat thevertexof a cubeor of a tetrahedron)
is thesumof thefaceangles
thatformit.Thusthemeasureof a vertexangleof a cubeis 2700, becauseeachof thethreeface
is 1800,
angleshas measure900, and themeasureof a vertexangleof a regulartetrahedron
becausethree600 anglessumto 1800. Now,he reasoned,
sinceeightcubesfillthespaceabouta
point,a necessary
and sufficient
conditionforspace-filling
is thatthesumof the
by tetrahedra
trihedral
at a pointbe equal to theproduct8 X 2700. Since12X 180= 8 X 270,it
anglesmeeting
followsthattwelveregulartetrahedra
fillthespaceabouta point,in agreement
withSimplicius.
Averroes's
theoryof anglemeasurewas generalized
by the 13thcentury
EnglishFranciscan
scholarRogerBacontoincludepolyhedral
anglesformed
byfourormoreplanes.In thecourseof
thatAristotle
this,Bacondiscovered
had missedsomething:
themeasureof a vertexangleof an
octahedron
is 4 X 600 = 240?,and 9 X 240= 8 X 270,whencenineoctahedrafillspace abouta
thisresultas a significant
advance.Buthe was awarethattherewas some
point!Baconregarded
in public"thattwenty
aboutthisquestionsincein Paris"a foolhad asserted
controversy
pyramids
fillthespace about a point.Bacon added thatto settlethematterit wouldbe necessaryto
ofwhatmenin thattimecould
understand
Euclid'sBookXIII. But,Struiksays,"It is indicative
andcouldnotdo, thattheypreferred
to Euclidor
lengthy
disputesto eithercalculating
according
a singlemodel."
takingthetroubleto construct
Actually
it is intuitively
reasonablethatthemeasureof a polyhedral
angleshouldbe, in some
way,closelyrelatedto thesumof thefaceanglescomprising
it,but therelationship
is not as
simpleas thesescholarssupposed(seebelow).Thatsomething
waswrongwiththeAverroes-Bacon
theorywas firstpointedout by the EnglishscholasticThomasBradwardinus
(1295-1349):if
filledthespaceabouta point,thentheywouldtogether
twelveregulartetrahedra
forma convex
withtwelveequilateral
polyhedron
triangular
faces,whichwouldbe a sixthregular
solid.Also,he
noted,thetheory
mustbe incorrect
becauseAristotle
didnotincludetheoctahedron
in hislistof
to Bradwardinus,
Aristotle's
space-fillers.
tetrahedra
couldbe obtainedbyjoiningthe
According
vertices
of a regular
icosahedron
to itscenter;in thiswaywe obtainthetwenty
tetrahedra
of the
Parisianfool.Bradwardinus
was unsurewhether
thesetetrahedra
wereregular.
(Theyarenot,but
unlikePotaman'stetrahedra
thisis notobvious:detailedcalculations
areneededto establishthe
fact that the ratio of the lengthof the icosahedraledge to the vertex-center
distanceis
likesomelatercommentators,
approximately
1.05.)PerhapsBradwardinus,
wouldhaveaccepted
of thesetetrahedra
on thegroundsthatAristotle
the nonregularity
did not explicitly
require
He appearsnotto haveinquiredwhether
regularity.
thispackingarrangement
couldbe repeated
to fillall of space(it cannot,sincetheicosahedron
is nota space-filler).
230
MATHEMATICSMAGAZINE
butit
achievement,
was a significant
theory
of theAverroes-Bacon
refutation
Bradwardinus's
by the
The regularsolidsare characterized
correct.
shouldbe pointedout thatit is notstrictly
thatthe same numberof faces meetat each vertex.If two regular
additionalrequirement
whosesix facesare equilateral
arejuxtaposedalonga face,we obtaina polyhedron
tetrahedra
convexpolyhedra
but it is not a regularsolid. There are fourother"irregular"
triangles,
onewith
including
triangles,
[3]) all ofwhosefacesareequilateral
called"deltahedra"
(sometimes
twelvefaces.
that nine
could have been extendedto provemorerigorously
argument
Bradwardinus's
octahedrado not fillthespace about a point.For if we cut theoctahedraby planespassing
withnine
at thepoint,we obtaina convexpolyhedron
oftheedgesmeeting
theendpoints
through
or nota given
anotherwayof decidingwhether
Thissuggests
squarefaces,whichis impossible.
edgee which
polyhedral
fillsthespaceabouta point:findtheshortest
ofpolyhedra
arrangement
meetsthepoint,thendrawa sphereaboutthepoint,choosingtheradiusr ofthesphereto satisfy
whoseedges
regions
polygonal
ofthesphereintospherical
r< e. In thiswaywe obtaina partition
faceswhichmeetat thepoint,and whosevertices
are thetraceson thesphereof thepolyhedral
are thepointsat whichthe spherecuts the edgeswhichbound them.Later,whenspherical
definedto be
anglewas correctly
themeasureof a polyhedral
had beendeveloped,
trigonometry
condition
and sufficient
polygonfoundin thisway.Thus a necessary
thearea of thespherical
to fillthespace abouta pointis thatthesumof theareasof thesepolygons
forthepolyhedra
equal thesurfaceareaof thesphere.
beginto be
whenEuclid was again studied,did the confusion
Onlyin the 15thcentury,
workon
(1436-1476),theauthorof an important
Muller,or Regiomontanus
resolved.Johannes
from
tell
as
we
can
in
critical
the
a
spirit,
discuss
to
problem
was thefirst
trigonometry,
spherical
bodies,thatareusuallycalledregular,
"On thefivelike-sided
thelengthy
titleof hismanuscript,
to thecommentator
and whichof themfilltheirnaturalplace,andwhichdo not,in contradiction
thisworkwas lost,but subsequentauthors,probably
Averroes."Unfortunately
on Aristotle,
by him,discussedtheproblemin a similarway,pointingout thatit is clearfrom
influenced
They
obtainedfromitarenotregular.
thatthetetrahedra
oftheicosahedron
Euclid'sconstruction
fillthespaceabouta
octahedraand eightregulartetrahedra
six regular
also notedthattogether
at a
meeting
thebases of theeighttetrahedra
in Potaman'sconstruction:
point(thisis implicit
and
the
cube
for
see
FIGURE
Except
4).
forma regularoctahedron;
cube vertextogether
packingwiththecube,thereare no otherwaysto fill
of thetetrahedra-octahedra
combinations
solids.
spacewithregular
fora longtimeafterwards;
errorinitsvariousguisespersisted
Aristotle's
Despitethiscriticism,
it by carelesslyacceptingthe earlier
who shouldhave knownbetterperpetrated
scientists
somescholarscontinued
admitted,
generally
Evenwhentheerrorwasfinally
arguments.
fallacious
The correct
requiredregularity.
on thegroundsthathe had not explicitly
to defendAristotle
polygonwas firstpublishedin a book by AlbertGirard,in
forthearea of a spherical
formula
J. Broscius(1591-1652)devoteda largeportionof an
1629.Later,the Polishmathematician
he
discussionof thisquestion.In thecourseof his argument
book to a thorough
important
best
is
this
book
is
that
the
it
for
and
of
a
the
area
polygon
for
spherical
developeda formula
and
spaceabouta pointwas discussedcorrectly
knowntoday.Hereat lasttheproblemof filling
in detail.
to experiment.
fromspeculation
Thisachievement
cameat thetimewhensciencewas turning
theproblemof
In thestudyofcrystals,
againbecamea focusof interest.
ofmatter
The structure
tookon a newsignificance.
polyhedra
spacewithcongruent
filling
revivetheproblem
?2. Crystallographers
Plato had not been concernedwiththeproblemof how externalformis achievedby the
centuriesthatwe findthe first
and seventeenth
stackingof particles.It is in the sixteenth
whoalso madean
Kepler(1571-1630),
Johannes
ofthisissue.Thegreatastronomer
investigations
in the
becameinterested
offilling
theplanewithpolygons,
to theproblem
contribution
important
VOL. 54, NO. 5, NOVEMBER 1981
231
(a)
(c)
(b)
(d)
spheresare arrangedat theverticesof cubes;each spheretouchessix others.
FIGURE 5. (a) In simplecubicpacking,
(b) In face-centered
cubicpacking,spheresare arrangedat theverticesand facecentersof cubes;each spheretouches
twelveothers.(c) Whenpacking(a) is compressed,
thespheresare deformed
intopolyhedra
withsix faces (cubes).
thespheresare deformed
dodecahedra).
intopolvhedra
withtwelvefaces(rhombic
(d) Whenpacking(b) is compressed,
causesof theformsof snowflakes
and wrotea bookletabouthisideasas a New Year'sgiftto a
arecomposedof minutespheresofice,he studied
friend
in 1611[9].Postulating
thatsnowflakes
formsthatspheres
forspheresand also someof thepolyhedral
variouspackingarrangements
In thisway,he discovered
wereuniformly
wouldassumeifthepackingarrangements
compressed.
severalspace-filling
For example,if thespheresare arrangedin whatis knownas
polyhedra.
formsare cubes; if theyare arrangedin theso-called
simplecubicpacking,thecompression
formsare rhombic
dodecahedra
(FIGURE5). It is
face-centered
cubicpacking,thecompression
we finda completely
newapproachto the
to notethatin Kepler'sworkon snowflakes
important
in thepreceding
sectionwere
All of theauthorswhoseworkwas described
space-filling
problem.
facesat
withfitting
matching
concerned
givenpolyhedra
together
locally-specifically,
principally
a vertex.(Kepleralso tookthisapproach,in his studyof planetilings.)The properties
of the
theselocalpackingsdo notseemto have
whichcouldbe generated
byextending
spatialpatterns
been consideredimportant
(indeed,as we have seen,some authorsdid not even investigate
Keplerobtainedfromsphereexisted).The compression
polyhedra
whether
an extendedpattern
weresolutionsof a globalproblem:whatkindsof polyhedra
pack together
packings,
however,
to therequirements
ofa givenrepeating
pattern?
according
debated.
had been revivedand was beingvigorously
By thistimethe atomichypothesis
was a popularapproachto thestudyof matter.In 1665theEnglishscientist
Sphere-packing
formscouldbe explainedby a few
RobertHooke statedthathe couldshowthatall crystalline
of spherical
atoms("had I timeand opportunity")
and gaveseveral
basic packingarrangements
as closelyas possible,stillleave gaps; the
examples.But spheres,evenwhenpackedtogether
difficulty
(althoughthe existenceof a vacuumhad been
vacuumproblemwas a persistent
but
in 1643).One wayto getaroundit was to assumethatatomsarenotspherical
demonstrated
ofthissortwas thatoftheItalianphysician
and
in form.
The first
theory
polyhedral
post-Platonic
232
MATHEMATICSMAGAZINE
of salts.
in thestructure
(1655-1710)whowas interested
DomenicoGuglielmini
mathematician
form
of a
have
the
of
each
the
atoms
and
of
salts,
types
Thereare,he said, fourprincipal
The basic saltsare
or an octahedron.
a cube,a hexagonalprism,a rhombohedron,
polyhedron:
oftheseatoms.We
bycombinations
ofatomsofa singleshape;othersaltsareformed
constructed
didnotsolve
formanyreasons,
atoms,whichwasattractive
ofpolyhedral
see herethatthetheory
do notfillspace-as Guglielmini
to solve,since(regular)octahedra
theproblemit was intended
was aware.
thatthe
werebasedon theassumption
theories
andpolyhedra-packing
Boththesphere-packing
despitethefactthat
regularity,
is theresultofsomesortofstructural
ofcrystals
geometry
external
(FIGURE 6). For manyyears
speciescan varygreatly
of a givenmineral
of thecrystals
theforms
it
was based. In about 1782,
which
on
atomic
hypothesis
as
the
thisviewwas as controversial
betweenthevarious
relationship
thatthereis a definite
empirically
it was discovered
however,
suggeststhattheexternal
of a species.This strongly
formsassumedby thecrystals
polyhedral
and characteristic.
fundamental
ofsomething
is a reflection
formofcrystals
Angles;
ofInterfacial
as theLaw ofConstancy
is knownto crystallographers
Thisrelationship
J.B. L. Romede l'Isle
it was statedin itsmostgeneralformin 1783by theFrenchmineralogist
can be grouped
way.The facesofa polyhedron
in thefollowing
(1736-1790).It can be described
of a given
the
faces
FIGURE
6,
saw
in
As
we
symmetry.
by
are
related
which
in families
together
absent,
or evenentirely
thanin another,
crystal
maybe largeror smallerin one individual
family
of thespecies.
anglesareexactlythesame,and arecharacteristic
butin everycase theinterfacial
offacescan be obtainedfroma single
species,thefamilies
Romepointedoutthatforeachcrystal
this.
its verticesand edges,and devoteda book to demonstrating
basic formby truncating
formanypeople.)
fascination
stillseemstoholda tremendous
(Truncation
and so
was premature
of crystals
aboutthestructure
Romehimself
believedthatspeculation
theFrenchabbe
ofhisworkwas leftto others.Not longafterward
thestudyof theimplications
theoryof
Rene JustHauy (1743-1827)revivedand expandedthepolyhedral
and mineralogist
science
of
crystallography.
the
modern
of
marks
beginning
the
his
work
structure;
crystal
Thebuildingblocksof
of Haiuy'stheory.
features
We willnotehereonlythechiefgeometrical
the
molecules"
(whicharenotnecessarily
as "crystal
whichcanbe regarded
arepolyhedra
crystals
or
tetrahedra,
same unitsas chemicalmolecules).These polyhedramay be parallelopipeds,
fora crystalspecies.The "nucleus"of a crystal
prisms:the typeis characteristic
triangular
of
growsby theaccretion
and thecrystal
groupedtogether,
consistsof severalof thesepolyhedra
at lower
Five crystalsof pyrite.The dodecahedron
AtlasderKristallformen.
FIGURI. 6. AdaptedfromGoldschmidt's
leftis notregular.
VOL. 54, NO. 5, NOVEMBER 1981
233
FIGURE
7. Hauy's construction
of pyritefromparallelopiped
blocks.
moreof them,nowgroupedtogether
to formparallelopipeds
(FIGURE 7). Somecrystal
facesare
thussmoothandothersstepped,
butthelatterappearsmoothtous becauseofthesubmicroscopic
sizeof thesteps.On thishypothesis
Hailywasable to explaintheformsthatcrystals
assume,and
whyotherforms(suchas theregular
icosahedron
and pentagonal
dodecahedron)
cannotexistin
the crystalworld.He was also able to givean explanationof certainphysicalproperties
of
suchas cleavage.
crystals,
Nevertheless,
Haiiy'stheory
wascontroversial.
His constructions
didnotalwaysagreeverywell
withmeasurement,
whichprovoked
severecriticism
of hiswork.The arguments
centered
on the
reality
ofhisbuilding
blocks.Onlylaterwas theirvalueas an abstract
modelunderstood.
Haiiy's
workled to themodemconceptthatperiodicity-regular
repetition
in all directions-isthe
fundamental
structural
characteristic
ofcrystals.
The firststepstowardthisabstraction
weretakenby theGermanphysicist
LudwigSeeberin
1824.Pointing
outthatHaiiy'stheory
couldnotexplaintheexpansion
and contraction
ofcrystals
withchangesof temperature,
he proposedreplacingtheparallelopiped
blocksby a systemof
theircenters
pointsrepresenting
ofgravity,
whichhe calledthespacelatfice.
Thismodelhasbeen
fruitful.
exceptionally
in theirsymmetry
Latticescandiffer
andin thewayinwhichthepointsarearranged.
(In 1849
AugustBravaisprovedthatthereareexactlyfourteen
types.)The pointsof a spacelatticecan be
represented
by theendpoints
ofvectorsof theformv = dx+ by+ cz, wherea, b, and c are three
If we calculateI612we obtaina
vectorsand x,y,z rangeovertheintegers.
linearly
independent
homogeneous
quadraticformin threevariables.Conversely,
each suchformrepresents
a space
lattice.But different
formscan represent
the same lattice:how can we tellwhichformsare
Thisquestionis closelyrelatedto thenumber-theoretic
of
equivalent?
problemof the"reduction"
thatis,theidentification
oftheforms
to
quadraticforms,
bytheparameters
d,b,E. It is ofinterest
us becausea majoradvancein the space-filling
fromthe reduction
problemcame indirectly
ofwhichSeeberhimself
was thefirst
to finda solution
problem,
(1830).Seeber'sworkwascorrect
but aesthetically
unappealing:it appearedto be unnecessarily
long and complicated.This
to simplify
efforts
prompted
byseveralmathematicians
it,and it was in thecourseof thisthatP.
Dirichlet
introduced
a construction
fortheregionof spacecloserto a givenlatticepointthanto
foreachpointof a spacelattice,we
anyother(FIGURE 8). If theDirichlet
regionis constructed
lieat latticepoints.
obtaina filling
ofspacebycongruent
convexpolyhedra
whosecenters
The polyhedra
whichformDirichletregionsof a space latticeare relatedto one anotherby
In general,
whichlead to spacefillings
are calledparallelotranslation.
polyhedra
by translation
hedra.Parallelohedra
are thebuildingblocksof spacelatticesand thusare clearlyimportant
for
theoretical
But formanyyearstherewerecontroversies
about theirphysical
crystallography.
Is crystalstructure
interpretation.
reallyperiodic?If so, is the block structure
reallyan apmodel?How are theblocksrelatedto chemicalstructure?
And so forth.
propriate
234
MATHEMATICSMAGAZINE
a
S
*
*
*
*
0
*
0
*
pointset,theDirichletregionof a givenpointP is theregionof space closerto P thanto
FIGURE 8. Givena discrete
line segments,
byjoiningP to each of theotherpointsby straight
anyotherpointof theset. It can be constructed
and finding
the smallestconvexregionboundedby thebisectors.The Dirichietregionsof a
bisectingthesegments,
gaps.
face-to-face
to fillspace without
whichfittogether
polyhedra
in parallelorientation
space latticeare congruent
(Here theDirichletregionsof someof thepointsof a planelatticeare shown.)
E. S. Fedorov(1853-1919)believedthatthe
and geometer
The greatRussiancrystallographer
can be partitioned
of chemicalmolecules,
containgroupings
intowhicha crystal
parallelohedra
in turnintocongruent
theparallelohedra
and thatby partitioning
regions(Potaman'sprinciple)
in detail,and
parallelohedra
we obtainthe truesubunitsof thecrystal.Fedorovinvestigated
can be classifiedinto five
provedthe remarkabletheoremthat the convexparallelohedra
a dodecahedron
topologicaltypes:thecube,thehexagonalprism,therhombicdodecahedron,
octahedron
(FIGURE9). Thiswas
and fourhexagonalfaces,and thetruncated
witheightrhombic
one.
andis stillthemostimportant
thefirst
ofspace-filling
polyhedra
generalresultin thetheory
results,in a book, An
Fedorovincludedthis theorem,along withmanyotherinteresting
publishedin
whichhe had begunat theage of 16. It was finally
StudyofFigures,
Introductory
evento readit,
refused
(P. Chebyshev
1885,afterbeingturneddownby severalmathematicians.
in thesequestions.")Fedorov'sbookhas
scienceis notinterested
remarking
that"contemporary
is contained
proofofhistheorem
buta simplified
neverbeentranslated
intoa western
language,
in [7].
FIGURE 9. Fedorov'sfivetopological
typesof parallelohedra.
VOL. 54, NO. 5, NOVEMBER 1981
235
0
0
A~~~~~~~~~~~~~
A
B
(a)
B
A
(b)
Let E be thecenterof thefaceA BCD
tetrahedra.
congruent
intotwenty-four
FIGURE 10. The cubecan be partitioned
A, B, C, D, and 0 to E
(b) Joining
A, B, C, and D to 0 fonnsa pyramid.
and 0 thecenterof thecube.(a) Joining
foreach of the cube faces,we
Repeatingthisconstruction
tetrahedra.
the pyramidintofourcongruent
partitions
tetrahedra.
twenty
congruent
obtaintheremaining
Although
theoryof crystalswas validatedby x-raytechniques.
In 1912 the space-lattice
thespace-filling
turnedoutto be incorrect,
in crystals
groupings
Fedorov'sviewof themolecular
Thus the space-filling
of crystalstructures.
modelcontinuesto provideusefulinterpretations
as well as by mathematicians.
problemis a subjectof activeresearchby crystallographers
theycan nowbe constructed
withtheaid ofcomputers
interest;
regionsareofparticular
Dirichlet
and so providesomeanswers
aretetrahedra,
setsofpoints.Someof theseregions
forcomplicated
to ourtitlequestion.
fillspace?
?3. Whichtetrahedra
We are back to our titlequestion,and in thissectionexaminewhatanswersare known.The
we haveoutlinedin the
used to searchforanswershavetheiroriginsin thehistory
techniques
a dissection
ofPotaman'stechnique:
Let us beginwithan examplereminiscent
previoussections.
If we join each vertexof a cube to its center,it is
tetrahedra.
of a cube-but intocongruent
intofourcongruent
Each pyramid
can be further
dissected
intosixcongruent
pyramids.
dissected
tetrahedra
by joiningeach of its verticesto the centerof its squareface.Thus we obtaina
tetrahedra
(FIGURE 10). Otherspace-filling
congruent
partitionof the cube into twenty-four
this one (FIGURE I1). In the searchfor
partitioning
can be foundby further
tetrahedra
sincethesmallest
forsuchtetrahedra,
it seemslogicalto beginsearching
polyhedra,
space-filling
is notsolved,norhas a comprehensive
Buteventoday,thisproblem
offacesareinvolved.
number
beendeveloped.
all suchtetrahedra
todiscover
andenumerate
technique
mustsatisfy
tetrahedra
It is important
to notethatthereis no a priorireasonwhyspace-filling
imposedby the authorsdiscussedin Section1 or the global
eitherthe local requirements
tetrahedra
of thosediscussedin Section2. We shallsee thatthereare space-filling
requirements
alongwholefaces;it is possiblethatthereare even space-filling
whichdo not pack together
into theunitsof a repeating
are not groupedtogether
in whichthe tetrahedra
arrangements
(orboth)makesthespace-filling
oflocalor globalrequirements
theimposition
However,
pattern.
techBradwardinus's
tractable(to someextent).Potaman'sprinciple,
problemmathematically
toolsavailableat thepresenttime.
arestilltheprincipal
construction
nique,and Dirichlet's
ifwe firstconsiderthevariouswaysin whicha tetrahedron
willbe simplified
Our discussion
if thereis at
is said to be symmetrical
figure
intosmallerones.A geometric
can be partitioned
it
leaves the
on
that
that
can
be
performed
motion
operation)
(or symmetry
least one rigid
(For example,in theplane,if a
appearance(and apparentposition)of the figureunchanged.
in itsnewpositionthefigure
appearsexactlyas it
is rotated180?aboutitscenter,
parallelogram
marksuchas a labelon a vertexwereadded
did in itsoriginalposition.Onlyifa distinguishing
236
MATHEMATICSMAGAZINE
~~~~~~~~0
0
E~~~~~~~~~~~~
A
B
F
(a)
A
A
O
F
(b)
A
F
(c)
FIGURE I 1. (a) The tetrahedron
of FIGURE 10,withF themidpoint
of edgeAB. If we taketheedge lengthA B to be
equal to 2, thenA E = BE = r2, OB = OA = V and OE = EF = AF = BF = 1. The tetrahedron
is symmetric
about
the plane through
E, 0, and F. (b) The plane through
E, 0, and F dividesthe tetrahedron
intotwo mirror-image
AEOF and BEOF (we showonlyAEOF). Each has a two-fold
tetrahedra,
rotationaxis-the axis of thetetrahedron
AEOF passesthrough
themidpoints
ofA0 and EF. (c) The tetrahedron
AEOF can be partitioned
intotwocongruent
in two ways:by a plane through
tetrahedra
A, 0, and the midpoint
of EF, and b) a plane throughE, F, and the
midpoint
ofA 0.
in a diagonalof a
had beenmoved.Reflection
couldyou tellwhether
or nottheparallelogram
A symmetrical
unlesstheparallelogram
is a rhombus.)
is nota symmetry
parallelogram
operation
intocongruent
objectcan be partitioned
partswhichthesymmetry
operationmapsonto one
thentheremaybe severalwaysto do this.The
another;if it has severalsymmetry
operations,
thatthesepartsmaybe chosento be tetrahedra.
The
has the unusualproperty
tetrahedron
is
of a tetrahedron
and thewaysin whichit can be partitioned
relationbetweenthesymmetry
in a planeor rotatory
is reflection
showninTABLE1.Noticethatiftheoperation
reflection
(i.e.,a
as describedin TABLE1), thenwe obtainmirror-image
rotationfollowedby a reflection
pairs
havemirror
in thesamewaythat
themselves
differ
which,unlessthesenewtetrahedra
symmetry,
do. Sincereflection
cannotactuallybe performed
in
coordinate
right-and left-handed
systems
is sometimes
called an "improper"motion.Some authors
three-dimensional
space,reflection
thatfillspacewith"properly"
betweentetrahedra
distinguish
congruent
copies,andthose,suchas
ofFIGURE11(b),whichmustbe accompanied
thetetrahedra
bytheirmirror
images.
The first
tetrahedra
was carriedoutby D. M. Y. Sommerville
systematic
studyofspace-filling
in manyfieldsof science.Accordingto The
(1879-1934),a geometerwithdeep interests
helda specialappealforhimandcrystal
forms
Dictionary
ofScientific
Biography,
"crystallography
of repetitive
doubtlessmotivatedhis investigation
space-filling
geometric
patterns."The imforhis studyof tetrahedra
mediateinspiration
was, however,an errormade by a student.
to
Sommerville
wrote,"In theanswerto thebook-work
question,set in a recentexamination
one candidatestatedthatthethreetetrahedra
intowhicha
thevolumeof a pyramid,
investigate
insteadof onlyequal in volume.It was an
triangular
prismcan be dividedare congruent,
in orderthatthe threetetrahedra
the conditions
shouldbe
interesting
questionto determine
whattetrahedra
to
the
wider
determine
can fillup spaceby
and
this
led
congruent,
problem-to
wrotetwopaperson thesubject.The firstdealtwith
repetitions"
[22](FIGURE12).Sommerville
withthepartition
of triangular
thewiderproblem;in thesecondhe was concerned
prismsinto
thesetetrahedra
couldfillall space.
andwhether
congruent
tetrahedra,
VOL. 54, NO. 5, NOVEMBER 1981
237
TABLE
of tetrahedra.
1. Tetrahedral
and partitions
symmetry
B
/
J
G
_ __
A
A
J
-
- - -
[ \
/
D
All tetrahedra
describedare represented
schematically
by thedrawing
at theleft.Verticesof thetetrahedra
are labeledas shown,withletters
A, B, C, and D. Midpointsof theedgesAB, CD, BC, AD, A C, and
BD, are denoted E, F, G, 11, I, and J, respectively.To show two edges
are thesamelength,
we markthemwiththesamesymbol(either/, /7,
or ///), as is customary
in elementary
geometry.
C'
1. Symmetry
2. A singletwo-fold
axis throughE and F.
(1800) rotation
The tetrahedron
can be partitionedinto two asymmetric
congruent
tetrahedra
in twoways,by plane ABF or plane CDE.
2. Symmetry
222. Threemutually
two-fold
perpendicular
axes, through
E and F, H and G, I and J. Each of theseaxes permitsa partition
of
thetetrahedron
intotwocongruent
tetrahedra
as describedin 1. Since
oppositeedgesare equal, each axis producesjust one partition.
3. Symmetry
m. A single mirrorplane. This tetrahedron
has two
scalene faces,ABC and BAD, and two isoscelesfaces,
mirror-image
BDC and ACD. A plane through
A, B, and F dividesthetetrahedron
intotwoasymmetric
tetrahedra.
mirror-image
2
238
---
---I
MG
MATHEMATICSMAGAZINE
TABLE
oftetrahedra.
andpartitions
1. Tetrahedral
symmetry
axis EF whichis the
has a two-fold
4. Symmetry
2m. This tetrahedron
of twoperpendicular
mirror
planesABF and ECD. It can
intersection
withsymmetry
be partitioned
in twoways intocongruenttetrahedra
+ ------
5. Symmetrv
class containsclasses 1-4: it has
4m. This symmetry
axes and twomirror
by reflection
planes.It is generated
threetwo-fold
in one of the mirrorplanes and by a four-fold
rotatory-reflection
(denotedby4): rotation900 aboutEF followed(nonstop)byreflection
to
perpendicular
in theplanethrough
thecenter0 of thetetrahedron
-
EF. This
operation maps the four faces of the tetrahedrononto one
anothercyclicly.Thus if we join the verticesto 0 we obtain four
withsymmetry
in: ABCO, BDCO, BADO, and
congruenttetrahedra
ACDO.
axis BQ, whereQ
has a three-fold
3m. This tetrahedron
6. Symmetry
it.
planespassingthrough
is thecenterof faceACD, and threemirror
can be partitionedinto six asymmetric
congruent
The tetrahedron
;
-/
Jt--
2
tetrahedra.
0'
Z
- -.
- - --- - 7axes.
VOL. 54, NO. 5, NOVEMBER 1981
All facesare equilateral.Each of EF, HG,
7. The regulartetrahedron.
and IJ is a 4 axis,and thereare six mirror
planes.The linesfromeach
(1200) rotation
vertexto thecenterof theoppositefaceare three-fold
The tetrahedron can be partitioned in all of the ways shown
above-and in otherwaysas well(thediscovery
of whichwe leave for
thereader).
239
A'
b
a
Al
d
_
LC
Cl
d
'a'
b' \
d
b
B
ABCB', B'A 'CA, andA'B'C'C.
intothreetetrahedra,
prismABCA'B'C' can be partitioned
FIGURE 12. A triangular
to be one which
tetrahedron
defineda space-filling
In thefirstpaper,Sommerville
copiessuchthat
congruent
(a) fillsspacewithproperly
arejuxtaposedface-to-face.
(b) thetetrahedra
or isosceles,thenit can be matchedto thecorresponding
is equilateral
If a faceof a tetrahedron
imagealso
it can be matchedonlyif its mirror
copy; otherwise
congruent
faceof a properly
With
symmetry).
itselfmusthavemirror
thetetrahedron
(consequently
appearson thetetrahedron
intotwokinds:
tetrahedra
classified
space-filling
and observations,
Sommerville
thesedefinitions
all of whose
mirror
without
symmetry
and (2) tetrahedra
withmirror
symmetry,
(1) tetrahedra
facesare isosceles(FIGURE 13).
First,he
thenaddressedtheproblemfromboththeglobaland local viewpoints.
Sommerville
kind(FIGURE
of thefirst
fourtetrahedra
to thecubeand discovered
appliedPotaman'sprinciple
can be partitioned
Sommerville
further,
as we haveseen,someof thesetetrahedra
14).Although,
condition
(a). He then
wouldnotsatisfy
tetrahedra
becausetheresulting
did notdo so, evidently
at a
of thefirstkindcan be fittedtogether
in general,thewaysin whichtetrahedra
considered,
thatthesetetrapatterns
thetriangular
he enumerated
technique,
vertex.UsingBradwardinus's
andconcludedthatthefourhe hadfoundtheother
ofa sphere,
hedrawoulddefineon thesurface
he did notcarryout a
reason,however,
wayweretheonlyonespossible.For someunexplained
is not
of thesecondkind.Thus his claimto completeness
fortetrahedra
similarenumeration
justified.
( I)
(2)
to fillspace
fora tetrahedron
exclusiverequirements
thereare twomutually
to Sommerville,
FIGURE 13. According
or (2) the
has mirror
symmetry,
(I) the tetrahedron
of properly
copies face-to-face:
congruent
by thejuxtaposition
butall itsfacesare isosceles.
tetrahedron
does nothavemirror
symmetrv
240
MATHEMATICSMAGAZINE
(ii)
(i)
(iii)
(iv)
tetrahedra.
(i) The firsttetrahedron
is thatof FIGURE 10. (ii) The
space-filling
FIGURE14. The fourSommerville
secondis foundbyjoiningtwoverticesof a cube whichsharea commonedgeto thecentersof twoadjacentcubes,as
of thefirsttype.)(iii) The third
is bisectedalongthecube face,we obtaintwotetrahedra
shown.(If thistetrahedron
a cube vertex,face
along a commonface through
tetrahedron
is obtainedfromthe firstbyjoiningtwotetrahedra
tetrahedra,
we can subdivideit intofourcongruent
centerand cube center.(iv) Since thesecondhas 4m symmetry,
as describedin TABLE1.
symmetry,
each of whichhas mirror
prism
intowhicha triangular
showedthatif thetetrahedra
In thesecondpaperSommerville
thena = b'= c' and one of theseadditional
(FIGURE 12) are congruent,
can be partitioned
amongtheedgesmusthold:
relations
(i) a'= b = c = d, 3a2 =4b2,
(ii) a'=b=c
a' = c = d,
(iii) a' = b= d orequivalently
(iv)b =c=d.
in FIGURE 15.
areshownschematically
bytheserelations
defined
ofprisms
Thefourfamilies
obtainedfrom
couldfillall of space.The tetrahedron
thesetetrahedra
He thenaskedwhether
(i) is thesecondof thefourthathe had foundin thefirstpaper,and he showedhowthe
family
(iii) and (iv),oneof thethree
In eachoffamilies
otherthreecouldbe derivedfromitbypartition.
is not
of space-filling
definition
his
so
of
the
other
two,
and
mirror
is
the
image
tetrahedra
arguedthat
Sommerville
congruent,
(ii) are properly
of family
thetetrahedra
Although
satisfied.
images.
theirmirror
cannotfillspacewithout
theprismsof thisfamily
X-aJX~~~~~~~
)
(
into
prismswhichcan be partitioned
fourfamiliesof triangular
FIGURE15. Schematicdrawingsof Sommerville's
theotherfamiliesare infinite.
Thereis onlyone memberof thefirstfamily(up to similarity);
tetrahedra.
congruent
(iii)(iv)
VOL. 54, NO. 5, NOVEMBER 1981
U
241
ways)
can be divided(in six different
A parallelopiped
triangularprisms.The two
into two mirror-image
in thecenter0: ifwe
prismsare relatedbyinversion
join anypointP of prismABDEFI to O and extend
thisline segmentby lengthIOPI, we findtheconfespondingpointP' of theprismBCDFGH.
C
B
FIGURI: 16.
A
"I
D
N
F
0
/
N
E
H
G
P
prismfillsspace.Theeasiestwayto see thisis to notethatanyparallelopiped
Everytriangular
any
into two mirror-image
triangular
prisms(FIGURE 16). Conversely,
can be partitioned
faceto form
alonga parallelogram
imageupside-down
prismcan bejoinedtoitsmirror
triangular
and
tetrahedra,
parallelopipeds,
a parallelopiped.
(ThismayhavebeenHaiuy'sreasonforchoosing
intoprismsand
a parallelopiped
units.)Ifwe partition
prismsforhisbasicpolyhedral
triangular
partsarejuxtaposed
theirconstituent
face-to-face,
thentetrahedra,
and stacktheparallelopipeds
withproperly
prismcan be juxtaposedface-to-face
withtheirmirrorimages.A triangular
facehasmirror
orifonedoesandtheother
symmetry
copiesonlyifeachparallelogram
congruent
do notsatisfy
eithercondition
secondfamily
twoaremirror
images.The prismsof Sommerville's
(M. Goldberghas pointed
and thismaybe whyhe did notconsiderthemto be truespace-fillers.
in a
congruent
copiesof thetetrahedra
can fillspacewithdirectly
out thatprismsof thisfamily
we obtain
copiesof sucha prismend-to-end
as shownon thecover.Stacking
pattern,
helix-like
Thentheinfinite
of whichare equilateraltriangles.
length,thecross-sections
prismsof infinite
is notface-to-face.)
as in FIGURE 3. Butthisspace-filling
prismscan be packedtogether
derivedfrom
thetetrahedra
seemsa littlecurious.In general,
argument
Evenso, Sommerville's
conditions
(1) or (2) forface-to-face
eitherofhisnecessary
theprismsoffamily
(ii) do notsatisfy
is of thesecondkind.When
But in thespecialcase whena = d, thetetrahedron
space filling.
are assembledintotriangular
prisms,theycannotfillspace without
copiesof thesetetrahedra
theirmirrorimages,but thisfactdoes not provethatthereis no way theycan do so. By
but this
dihedralanglesone can showthatin facttheydo notfillspaceface-to-face,
calculating
leftunresolved,
thestatusof thetetrahedra
of thiskind
raisesagainthequestionSommerville
prism.
whichcannotbe obtainedfroma triangular
On theotherhandit is remarkable
that,as faras we are aware,all theknownspace-filling
of thetechnique
used to findthem,can be obtainedfromSommerville's
tetrahedra,
regardless
whichgeneratespace-filling
fourprismfamilies.H. S. M. Coxeterdiscussedthreetetrahedra
firstand second,
in theirfaces[15,p. 84]; theseturnout to be Sommerville's
copiesbyreflection
thetetrahedra
of
of thefirstshownin FIGURE 11(b). H. L. Daviesrediscovered
and thepartition
fourth
andobtaineda secondfamily
[16].He also showed
bypartition
.Sommerville's
prismfamily
edgeand
can be derivedfromthesebyspecializing
and fourth
tetrahedra
first
howSommerville's
discovered
thelatter.L. Baumgartner
and foundanotherby partitioning
angularrelationships,
andan additional
one obtainedfromthesecond
tetrahedra
Sommerville's
second,andfourth
first,
itwitha planecontaining
a two-fold
rotation
axis[13],[14].M. Goldberg,
restricting
bybisecting
secondfamilyin the two
Sommerville's
tetrahedra,
partitioned
himselfto properly
congruent
are not
[18]; as we havealreadynoted,thesespace-fillings
possiblewaysto obtainthreefamilies
foundby E. Koch in her computerstudyof a class of
face-to-face.
The five tetrahedra
tetrahedra
plusthe
Dirichlet
regions[20]are thefourSommerville
important
crystallographically
thereareanytetrahedral
thatcannotbe
Whether
space-fillers
foundbyBaumgartner.
tetrahedron
a triangular
obtainedbypartitioning
prismremainsan openquestion.
can be obtainedbypartitionwe can askwhether
Moregenerally,
space-filler
everytetrahedral
at thebeginning
of thissection,
whether
We can also ask,as we suggested
inga parallelohedron.
whichfillspace in an irregular
way. (Indeed,it is possiblethatsuch
thereexisttetrahedra
conditions
Sommerville's
(a) and (b).) More than2300 yearsafter
tetrahedra
mightevensatisfy
fillspaceand whichdo notis stillunresolved!
thequestionofwhichtetrahedra
Aristotle,
The generalspace-filling
open problemsabound;the
problemis stillwideopen.Challenging
and otherfields
butalso forcrystallography
notonlyformathematics
answerswillbe important
242
MATHEMATICSMAGAZINE
exceptforthe
of space.We do notknowtheshapesof space-fillers,
withthepartition
concerned
numberof
and certainotherspecialclasses;we do notevenknowthemaximum
parallelohedra
can have,thoughthenumberhas beenprovedto be finiteforone important
facesa space-filler
is thirty-eight,
of facesknownto occurin a convexspace-filler
generalclass.Thelargestnumber
thereis the
P. Engel[17]. Even moregenerally,
foundby thecrystallographer
as was recently
In Plato'swords,"their
spacewithcopiesof twoor morekindsof polyhedra.
problemof filling
whichanyone
andwitheachothergiverisetoendlesscomplexities,
withthemselves
combinations
mustsurvey."
whois to givea likelyaccountofreality
I wouldliketo thankBranko(irunbaum,Susan Petrelli(SmithCollege,'82) and LesterSenechalfortheirhelpful
versionof thispaper,and Deedie Steele(HampshireCollege,'81) forconstructing
commentson the preliminary
tetrahedra.
excellentmodelsof theSommerville
References
comments.
readingare providedforeach section,alongwithappropriate
forfurther
Suggestions
?1.
[ ]
[2]
[3
[4 ]
[ 5]
?2.
[6 ]
loci,NieuwArchiefvoorWiskunde,2nd series,15(1925) 121-134.(Despite its
DirkJ.Struik,De impletione
by thepresentauthorcan be obtainedfrom
in Dutch; an Englishtranslation
Latintitle,thearticleis written
MassachusettsInstituteof Technology,Cambridge,MA
heror Prof.Struik,Departmentof Mathematics,
omittedin our exposition.)
references
02138. Struik'sarticlecontainsmanydetailsand bibliographic
Dover,New York, 1948.(GeneralbackgroundinformaDirk J. Struik,A ConciseHistoryof Mathematics,
in thisearlyperiod.)
tionabout mathematics
A. Beck,M. Bleicher,D. Crowe,ExcursionsintoMathematics,
Worth,New York, 1969,pp. 21-30.
B. Grunbaumand (i. S. Shephard,Isohedraltilingsof theplane by polygons,Comm.Math.Helv.,53(1978)
542-571. (A detaileddiscussionof manypolygonswhichfilltheplane in additionto thethreeregularones
citedhere.)
pentagons,thisMAGAZINE, 51(1978) 29-44. (Discusses
Tilingtheplane withcongruent
D. Schattschneider,
convexpentagons.)
irregular
thespecialunsolvedproblemof tilingtheplane withcongruent
JohnC. Burke,Originsof theScienceof Crystals,Univ.of CaliforniaPress,Berkeleyand Los Angeles,1966.
up to about 1850.)
(Describesthehistoryof crystallography
L. Fejes T6th,RegularFigures,PergamonPress,New York, 1965,pp. 114-119, 121-123.
[7
[ 8 ] Alan Holdenand PhyllisSinger,Crystalsand CrystalGrowing,ScienceStudySeries,AnchorBooks,Garden
City,NY, 1960.(A veryreadableaccountof crystalsand theirproperties.)
OxfordUniv.Press,1966.
Latintextand Englishtranslation,
Snowflake,
[ 9 ] JohannesKepler,The Six-Cornered
to appear in A HistoricalAtlas of
[101 Marjorie Senechal,A briefhistoryof geometriccrystallography,
of
Unionof Crystallography.
(Describesthehistory
J.Lima de Faria (Editor),International
Crystallography,
up to thepresent.)
crystallography
geometric
in NatureIs a StrategyforDesign,MIT Press,1978.
[11] PeterPearce,Structure
Foundationsof Natural Structure:A Source Book of Design, 2nd ed.,
[12] RobertWilliams,The (Geometric
polyhedraare discussed;
Dover,New York, 1979.(The mostdetailedof severalbooks in whichspace-filling
anotheris [11].)
?3.
Funfzelle,Mathematische-Physikal
Raumesin kongruente
Zerlegungdes vierdimensionalen
[131 L. Baumgartner,
15(1968)76-86.
Semesterberichte,
Raumes in kongruenteSimplexe,MathematischeNachrichten,
_, Zerlegungdes n-dimensionalen
[14] _
48(1971) 213-224.
[15] H. S. M. Coxeter,RegularPolytopes,3rded., Dover,New York, 1973.
Proceedingsof the Colloquiumon Convexity,
[16] H. L. Davies, Packingsof sphericaltrianglesand tetrahedra,
Copenhagen,1965,pp. 42-51.
Z. Krist.,154(1981)199-215.
von kubischerSymmetrie,
[17] P. Engel,Uber Wirkungsbereichsteilungen
J.Combin.Theory,16(1974)348-354.
space-fillers,
familiesof tetrahedral
[18] M. (ioldberg,Threeinfinite
tiles,Bull.Amer.Math.Soc., New Series,3(1980)
[19] B. Griinbaumand Gi.S. Shephard,Tilingswithcongruent
shapes.)
951-973. (Gives a surveyof thegeneralproblemof fillingspace withcongruent
mit
zu kubischenGitterkomplexen
und Wirkungsbereichsteilungen
[20] E. Koch, Wirkungsbereichspolyeder
Marburg/Lahn,1972.
Philipps-Universitat,
Dissertation,
wenigerals dreiFreiheitsgraden,
Proc. Royal Societyof
Divisionof space by congruenttrianglesand tetrahedra,
[21] D. M. Y. Sommerville,
43(1923) 85-116.
Edinburgh,
in Euclideanspace,Proc.EdinburghMath. Soc., 41(1923) 49-57.
tetrahedra
Space-filling
'_,
[221
VOL. 54, NO. 5, NOVEMBER 1981
243