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The effect of cell division on the shape and size of hexagonal cells.

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“ L ’avenir de l’histologie et sp&cialenient de la cytologie est tout
enticr dans la niicrochimie. ”-Opinion prt‘vulontt .
Division of tctrakaidecalietlral cells, when ai*ranged with
hexagonal siirfaces above aiid below, OCCUI‘S in the equatorial
plane, a i d , perpendicular to that plane, in three vertical
planes a t angles of 60” with each 0ther.l 111 a simple epitlielium the ecyuat orial divisions are eliminated. If they occurred they would start stratification. There remain, then,
0111~7the three vertical plaiies sliown in fignrc 1 a t a, b, aiid c ;
and the effect of division in these plaues upoii the shape and
size of cells in simple epithelium is the primary object of
this report.
If the ontline of the cell is a regular hexagon, with sides
of unit Ieiigth, its perimeter measures 6 units and its area
2.?598sq. units. Division in the plane a, 6 , or c, Tvliich bisects
the area, atlds t o the perimeter; each resultiiig cell is a very
irregular pentagon with x perimeter of 4.732 for a n area of
1.299. 111 cells with plastic W W ~ Isnrfnce
tciision promptly
rcdnces the rclatively great length of this perimeter by making the pentagons as regular a s possihle. If the area remains the same, the pentagon can be made regular by shortciiiiig its sidm to a iuiiform length of 0.869. But this would
mmii a ~ ~ o i . I ’ ~ ~ s i ) o irecliiction
~ ( l i ~ i g in the coiitiguous sides of
Thc tl11 isioir of tetl.:tk:iitlr(’;ihrdral ~ c ~ l lhsa q alread)
An1c.r. Acnd., 1923, \n1. j8, p. ii-$(i
ct s i q .
1, J t ? ( O R I )
TI \ I B F 1 <
I 1 SO
ati?joirlilig (.ells, and a consequent increase of their areas relas. To meet this situation the peiitagons
their sides of nearly unit lengtli. If absolilt ely regular arid of unit sides, their t t r ~ a s would he&
iiicrcasccl by 32.4 per cent arid would become 1.720. Tliejthen would l i a v ~ relatively,
the shortest perimc.tcrs possiblc
f o r (~111sof peiitagoiial form. If they grew still larger in aii
clffort to attain the size of the parent cell, it would aff’ect
1111l’i~vorablythe iteiglilmi~iiig cells by makiiig their prrimtltc1r.s exceed the minima. for the arcas cnclosed. (’011scqueritly growth may he expected to stop a t that point,
rriakiiig tlic end result of tlic division of ilic hexagon the p * o tIuc*tiori of a pair of iicarlj7 regular pentagons of nearly miit
siclcs. ‘l’hcp arc small cells, with a comhiried area seldom
of t h e 1)ureiit wll ;
attaiiiiiig a 32.4 pcr cont iiicrensc o v ~ that
t1ic.y by n o mtwis double its sizc.
I)ivisiori of a hexagonal (bell in the plane tl of figure 1 \vould
i i o t lie expected f o r several re‘asoiis. It ~ v o u l dplacc tlic
p o l c n of tlitl mitotic spindle in a short axis of the cell. With
t l 1 ~adjoining cells it would give rise to unstable tetrahedral
i~iiglcsat either end. It would be contrary t o Errera’s rulc
fliiit the ‘incipient partition-wall of a dividing cc.11 tciiiis
t o l)e such that its area is the least p o s s i l k hy which the
(*mi 1)e e~iclosccl.’ ’ 2
‘I’etrahcdral aiiglcs a r e
i i o t aiikiio~viiin tissues: the mitotic spindle by a skewiiig of
tlic ~ i ~ ~ i s i o i i - r ) lmay
~ ~ i t occupy
a short axis of the cell ; and
F:rrcra’s law may hc flagrantly evaded, as is regularly tllc
ciise in the cambial cclls ~vlii(*IiBailey studied,3 where (.Iongated cells split Icngthwise. Nevertheless, if thc axes of a
( ~ 1 1 1a1.e equal arid its outline is a regular hexagon, there are
i cliaiices f o r division iii any one of the planes a, h, and c,
with the probabilities against such planes as d.
I f in a sheet of liexagonal cells, a single one divides in
f lic plane a, l), o r c, it will protlucc n pair of pentagonal cells
‘Sir Il’Arcy Thompson: Growth a n d Form, ILIli, p. 318.
IZnlle\, I. W. The c:ririhium, etc., Am. .JouIn. Rot., 1900, x o l . 7, p1). 417-4.31;
J ~ i i r i i .( k n . 1’11) siol., 1920, xol. 2, pp. 51!)-5:13.
aiid ~ v i l canse
t v o a d j o i i ~ h gcells to become heptagoiial (fig.
3 ) . If thc six cells surrouiidiiig the dividing cell should also
divide in the plmes u, b , anti +-two in each plane and ZITraiigecl as in figure 4-there ~wmldbe altogcther eight new
Fig 5
E'igs. 1 t o 1 1)i:igranis illustrating tlw r'fcct ~b w11 division on liesagonal
ecI Is : equal nuiiibers of pmtagoas and heptagons are 1mduwd, a n oetagon
sonictimcs replacing two heptagons.
Fig. 5 A binuclcate dodecagonal cell from t h e pig~ncnted layer of t h r retina
of n human tmibryu of S f months. x 700 dialn. All cells other
are miarkcd w i t h figurcs i n d i c a t i n g t h e i r nunrher of sides. The :I
31 cells sho\vii is exactly F.
liexagoiiul aild six lie\\- pelltagoiial cells ; two of the NITrouiidiiig cells woiild be made heptagonal and two octagonal.
The average number of sides f o r these six pentagons, two
heptagons, and two ociagons remains exactly six. Extending this procedure so that the 36 cells s ~ ~ r r o i i i ~ lthe
i n g cen-
tral one divide (and t.wclvc? of them in eac.11 of tlic three
t!lierc may be produced 43 iiex liesagoiis
plaiies a, $1, aiid
a11d 22 of pentagons and heptagons. ‘I.11~planes in the
t.c?sttlmployeci were plalced a t ra.iidom, except that the plane
i u any cell was iiever made the exteiision through i.t of the
l . ) I i t l i ~clioseii
in ail ad,joi.ning cell. I n tissues t.lierc is evidcncc. tlia t. t,he ext.eiision of a division-plane through. ail adjoiiiiiig cell is not always avoided. Some irregularity of thcb
t~t!,or sliglit obliquity of tlic pl.anes, usually prevcti1t.s the
formation of t.he tetrahedral aiigles which are iiievi.t.ablc i.n
a tliagram. But the cliagrams reveal the importa1i.t facts that.
division of bhe hexagons produces mi equal number of pentagons aiid. heptagons--Hii oct,agoii somct.imcs replacing two
.heptagons; aiid that., as division proceeds, both pentagons a i d
heptagons are being restored to the prima.ry liexagoiial form.
l’hough proba.bly never rea.lizcd in tissues, the a.rra.ngemeiit
of‘ planes in figure 2 permit,s restoration of regulai: liexagond
oiitliiies throughout.
(.!ells with more t1ia.n eight. or less t.haii five sides are not
iincommon. A n undivided hexagonal cell, if surrounded by
six w h i ~ hbisected radially, would acquire twelve sides. It
wonld ht? enclosed by twelve pent,agoiis, which would have
made six periphera.1 cells heptagons. Again the t.ottil loss
and gain hi sides is equal-one dodecagon, six heptagons and
t\velve pent.agoiis averaging si.x sides apiece. I n the pigmeiit
layer of the retina I caii recwrd a dodecagon less simple in
origin (fig. 5). .It is surrounded by seven hexagons, .fonr
~)cntagoas,a.iid a quadrilat.era1. T h e excess of six sides in t.he
c?t.iitralcell is met by R deficiency of six in the cells in coiltact
wit,li it.; and the second circ,le of investing cclls-ten hcsi~goiis,
four pcliit.agons mid four hepta.gons-is another iiistaiice of
thc! prcvailiiig baltmce. .Dodecagons in the retina have h e e ~
iiot~etlby Rouin,-’ and possibly cells with eveii more sides
OCCUI’, sirice oiie of the bordering cells in 0111. case h a s two
nnclei, incomp1et.e division. F o r less t.han fire
sides a hexagon presumably divides uneyuall\-. The divisioli
’ Priauaiit ct Houim, Trait6 cl’histologie, T. 2, 191.1, 1). 6 i 4 .
of a peiitagoii is an altcriiative explanation. I n oiie or the
other of these ways doubtless most of the quadrilaterals are
produced, aiid they a r e not rare, though triangular cells a r e
almost uiikiio.v\.ii. Thus cells with more than seven or less
than five sides a r e iii different categories. The former arise
from exceptioiial groupiiigs of normal divisions ; hut tlie latter
cannot he interprotctl in that wi;\-.
The maiiiicr iii which pentagons are produc~?daiid why they
a r e small havc already been explained. The hcptagons have
becii S ~ J O W T Hto be merely Iievagoiial cells along one side of
which cell division has taken place. A t first tlieir perimeters
measure six units aiicl their areas a r e those of regular hexagons-2.598.
If, Iiowever, these cells become regular l i e p
tagons without inci.easc in area, each side is reduced t o 0.846 :
should these sides be madc of unit leiigth, the cell must eiilarge hy 39.9 per cent aiid acquire an area of 3.633..
I n a simple epithelium in \vliicli the cells, after dividing,
return to their original size, the elid result of a single clivisioii
of every cell is clearly both to double the area of the layer
a s a whole, aiid to produce from every hexagon another of
equal size. For geometrical reasons it happens, curiously,
that when the division of R single cell (as in fig. 3 ) introduces
a iiew element, although the total increase is closely equiralent to adding a new hexagon of staiidard size, only oiie third
of this increment is provided hy the two pciitagonal daughtercells-two thirds come from the formation of tlie two heptagons. Considering tlie cells as regular polygons, three hexagons, instead of adding a fourth and maliing their combined
area 4 X 2.598, o r 10.4, become resolved into two pentaGoons
arid two heptagons with a total area of 10.7-that
2 X 1.720 2 X 3.639. But i t is impossible to bring together
regular polygoiis of these shapes without leaving interstices.
Tlie problem of correcting the decimals to fit the actual conditions should interest some mathematician of leisure. The extraordinary liiological conclusion is, that ~-1ieiia cell with
plastic walls receives a new side tlirough division of ail adjoining cell, it gro~7sas it never would liave done had i t s
ii(ligli1)or rc~rriaiiicclclniescctit. TVlic~na cell divides beside a
i~.gularpciitagoii, Tvliicli tliei~eafterbecomes a rcgiilar liesagon of tlic original lciigtli of side, the growth of the pentagoil is 51 1wr t w i l !
‘I’llc. gro\vll1 w11ich lras bccw : igiietl t o thc~several forms
of’ pol?-gons depends npon tlirir acquisition of sides of equal
l ~ ~ g t lxit, the same 1 ime 1)ecomiiig regular aiid of minimal
pcirimctcr. The cAx-tcliit to n.1iic.h this is true of ichtual cells can
l x ~mcasnrccl clii*ectIy. All thts x alls of thcb cclls ontlincd in
figurc 1 2 (p.31.5)wero mcasured in a convenieiit pliotographic
tiiilargement. rl’heir average length mas made tlie liiiear unit,
~ i i dits square tlie uiiil of surface. The area of every cell was
measured wit11 a plaiiirrieter, aiid tlie resnlts a r c presented in
t21lde 1.
IVe have assumctl that the pentagons arise from tlic b i s w
tioii of liesagoris of average size, wliich means that at the
time of thoii* oi.igin tlicir areas a r e 1.3 and tlicir sides arcrage 0.946; and that tlicy grow until nearly regular mid of unit
sitlc, tfins iiicreasing by 32 per cent. ‘L’he seven pentagons
in figure 12 have grown only 15 per cent,; they are still small,
hi1 t Iinvc become approximately regular f o r their short pclrimeters as sliowii in tlie last column of the table. The four l i e p
t agoiis, ho~vever,have exceeded the expected growth of 40
per cent, being 4(i per (wit larger than the hexagons, and having sides of more th;m unit length. The small size of the
ptiitagoiis and large size of the heptagons in this little group
is ilouhtless a chance occurrc’iice. By including more cells-
making fifty in all-the average length of side of the pentagons is raised to 0.996, and that of the hexagons a i d heptagons lowered to 1.006 and 1.009, respectively, indicating
that i n general, the average length of side for these three
kinds of polygons is the same. Within this average the range
is very great-from 0.5 to 1.5-and extremes may be combined
in a single perimeter; yet, as shown in the last column of the
table, surface tension brings order out of these complcxities
by making every perimeter enclose more than 92 per cent
of t,he area wliich can possibly be brought within its bounds.
The average f o r the twenty-four cells is 96.6 per cent. Since
regular polygons caiinot be fitted together, 100 per cent is
unattainable, so that the actual percentages a r e distinctly
nearer mathematical perfection than they appear. The a p
plication of the purely geometrical coiiclusions of our introductory paragraphs to cellular tissue seems therefore fully
Figure 12, which we have examined in some detail, is from
a valuable paper by Dr. Georg Wetzel--‘ ‘Zur entwicklungsmechanischen Analyse des einf achen pri smatischeii Epithels”5-kindly brought to my attention by my associate, Mr.
Weatherford. It is a study of the cell outlines in the pigmented layer of the retina-‘ein
zierliches Jlosaik’--u-here
cells, as Jones described them i n ’33, a r c “very minute plates
of a n hexagonal form, accurately joined by their edge^."^
Covering much of the approximately spherical surface of the
eyeball, this layer grows a s nearly uniformly in all directions
as perhaps any layer in the body, and therefore is especially
favorable for this investigation. Here, as Thompson remarks, “conditions all point and lead to a perfectly symmetrical result. ”
Wetzcl counted the number of sides presented by the pigment cells in seven fields of the microscope, and found that
the various polygons occurred a s in table 2.
‘Arch. f . Entw. uiechxnik, 1926, U d . 107, pp. 177-183.
‘Thomas Wh:irton Junes: Yotire relatirc t o thc pigmmtum nigruin of t h e
eye. Edinburgh Jled. and Surg. Jourii., 1833, 101. 40, p. 81.
He is impressed with the fact that pentagons and heptagons
are present in nearly equal numbers,-in some fields exactly
even; but in seeking the cause, he does not so much as mention the partitioning effects of cell division, and thus overlooks the simple explanation which accounts f o r the case in
hand. Wetzel stresses the geometrical proposition that if a
plane is to be covered completely with approximately regular
polygons with three sides always meeting at a point, and if
with hexagons only pentagons and heptagons are present,
then the two latter must occur in equal numbers, however
they arise. This is important,-more so than the mathematical formula to express it, supplied him by Reinhardt. As a
student of topology, Reinhardt had published his inaugural
dissertation “Uber die Zerlegung der Ebene in Polygone,”7
but that he can write also simple equations will appear presently. Let n8,n4,etc. represent the number of octagons, quadrilaterals, etc. which are present, the number of hexagons
being omitted as variable and immaterial ; then Reinhardt ’s
formula “welche alle Falle umfasst,” as Wetzel expresses it,
is as follows:
2n8 m7 =n5 2n4
[To include nonagons, should it not be,-
+ 2x8 +
=‘ n 5
+ 2%,
Wetzel, verifying the formula as to equal numbers of
pentagons and heptagons in the retina, does not consider it
profitable to test it further, merely noting that in his preparations the number of quadrilaterals and octagons is very small.
Disregarding them, he counted the sides of 456 pigment
cells-hexagons, heptagons, and pentagons-for
which the
average number of sides proves to be 5.98. Including the two
quadrilaterals, three octagons and one nonagon which he
would disregard, the average becomes 5.99. If we consider
the polygons as independent of each other, discounting the
fact that they have sides in common, then for mathematically
’ 1naug.-Diss.,
Frankfurt a.M., 1918, pp. 1-85.
perfect results tlie 462 cells should have liad three more sides
than were found-2772 instead of 2769. This is perhaps tlie
most perfect demonstration of tlie geometrical form of cells
ever recorded. Greater pi-ecisioii is liardly expected in the
realm of pare physics.*
Iiisteacl of seeing iii the large size of the many-sidcd cells
ail effect of the iiumber of sides assigned to them, arid vice
versa with the small cells, which is our view of tlie matter,
TVctzel takes the reverse position. Any vital activity which
makes some of the cells large thereby increases their number
of sides, lie suggests. “And these larger cells, howwrer they
may liave arisen, must occasion in their neighborhood a rearrangement of cell forms, with tlie production of a coi-respoiitling iiumher of cells with fewer sides. ” But G raper
recogiiizes that a hexagonal cell “must hecome very large
t o reach a single additional cell ;” and later lie writes,-“it
must double its size t o come into relation with a seventh cell,
and trehlc it for contact with an eighth.”9 It is not clear
how those proportions mere cletermiiied, whicli certainly differ from the expected and actnal dimensions of such cclls, a s
shown in table 1.
What Wetzel descarihes in this general way may actually
take place in stratified epithelium. A hexagon may acquire
tTnirnprcssed W i t h his omn fiiidings, \vetzcl turns to R L(shrbuch-figure of the
r e t i n a l ctslli;, f r o m which he derires t h e shapes of 29 morcz to add to 111s list.
Thcse do not better the a i c m g e , and i i s there stwris to be n o r(’ason f o r including
them, wc have disregnrrlrd them altogether.
G I aper, L u h i g . Eiiie iieuc Arisc~lrnuangiilier pl~ysiologisclicZ c 1 l : ~ u s s c l i ~ l t u i i ~ .
Arch. f . Zellforscliung, 1914, Btl. 12, p. 387. Mec.lianische Eetrac~litungc~n
Vcrsuthc uhrr Zcllforrri und Z ~ l l g r o s s e . Arch. f . Entw.-mcchanlk, 1919, Bd. 45,
p. 44i.
a scveiitli side 1):- ciilargemcnt, as in figure 6A n may meet 6,
rnakiiig both of them hcytagons, and extending between c ant1
Snch a group, in figure
ti' \rliicli : ~ J X ' 1.cniierct1 p a n t a g o i d .
613, has been carefiilly outliiied from an actual specimen:
o x + h of the four (~11sw a s sectioned through its nucleus. It
niakcs a clefinite pattern, hnt one nol characteristic of the
retina, and iii fact not ahunclant in the stratified epithelium.
For the pentagonal ccllls do not remain quiescent. They, t 00,
enlarge, wedging their way Iatcrally, and hecoming hcxagoiial as the\- deprive the heptagons of their extra sides and
lift the overlying strata.l" That this has not occurred in the
gronp in figure 6 is perhaps clue to the small size of the
pciitagons, which a r e but one-half the area of the heptagons,
Fig. 6 1)iagran~ ( A ) a n d canicrit outline ( K ) of f o u r cells of 1ium:in oral
r p i t l i c ~ l ~ u i ~F~r.o m a n cml)ryo o f 5 inonthx.
X &SO cliain. Extensio~i of c.clls
n :ind b until the! nicet produrcs two Iieptagons arid two pentagons.
aiitl so happcn to be of the proportion which Griiper specified.
If tlie retinal hexagons are made pentagons by the enlargemcnt of acl jacent cells which become heptagons, then the pentagons should have tlie area of the hexagons. They ~voulcl
iiot he reduced in size hy this process. The fact that the
retiiial pentagons are smaller tliaii the hexagons, and the
ahserice of tlie characteristic grouping, lead to tlie rejection
of Wetzel's hypotliesis of uriequal gromf h. I t explaiiis
rncircly the Tariation in size of polygons of a given iium1)er of
sidcs .
Another way of accouiitiiig for the equal number of heptagous a r i d pentagons is based upon Griiper 's studies of cell
l') 'Phis is furtlicr t1cwrit)ctl 111 t h c €'roc.. Ainer. Arad., 1925, x o l . 61, pl). 30, 31.
' 1 ' 1 1 ~ ct,lls srctionc~(l111 figure (i arc not piiwvi, but tctrak:i~tiec~nhrtl~n.
fusion. His premise is that in the adult, when an organ invested by a simple epithelium has ceased to grow and neither
enlarges nor diminishes in bulli, there must be a cell lost for
every cell added through mitosis, provided the size of the
cells remains constant.ll I n regressive orgaris, like the yolksac of the dogfish, which he studied particularly, elimination
of cells outstrips production. H e fouiid that this elimination
occurred through fusion of hexagonal cells in a strictly simple
epithelium, and very interestingly he has described its effects
upon the cell pattern. In a hexagoiial mosaic he points out
that two adjacent cells a r e always enclosed by eight others,
arid should the two fuse and become one binucleate cell, they
would still be surrounded hy the eight; whereas should the
binucleate condition arise from the division of a nucleus in a
single cell, that cell would be surrounded by six only.12 One
important feature he omits to mention, namely that with the
formation of this large octagonal cell hy fusion (cell a,fig. 7 ) ,
two adjacent hexagonal cells become pentagoiis (fig. 7, h and
c). I n fact, erroneously in his diagram (which is here reproduced in fig. 8) these cells a r e shown as hexagons. It happens,
then, that in cell fusion, which is the reverse of cell division,
the total loss arid gain in sides is also equal. This outcome,
to which apparently Griiper nowhere refers, is further illustrated in the following arrangements.
If the fused octagonal cell becomes reduced to normal size,
Graper shows (fig. 8) that i t may become hexagonal, but will
then be in relation with two more pentagons, sometimes at
opposite poles, as in figure 8 at d and e. But these join the
group at unstable tetrahedral angles. Should cell f in figure
8 acquire contact with 9, and cell h with i, those hexagons
mould become heptagons-the
unstable angles would be
avoided-and the elid result of this symmetrical fusion would
be the loss of one hexagon arid the formation of four pentagons and four heptagons.
See footnote 9, page 339.
This assunirs that tlic nuclear division occurs in a hexagonal cell. Griiiper does
not mention that if nuclear dirision takes place in a n octagonal cell, the conditions
which he ascribes t o fusion would he duplicated.
T H E A N A T O M I C A L RECORD, VOL. 3 3 , N O . 5
A variant of the pattern shown in figure 7 is presented in
our figure 9, in which the two hexagons are considered to have
fused asymmetrically, that on the left draining, as it were,
the substance of the one on the right, and so forming the
irregular octagon a. Should cell a in figure 9 shrink still
further, it might become one of the six approximately pentagonal elements in the ‘rosettes’ (fig. 10) which Graper both
observed in tissues and interpreted as evidence of fusion.
The same thing would happen if any cell in the hexagonal
mosaic should be forced up and out, and the surrounding six
should come together below it, as in forming a stratified
epithelium. But the cells do not quite meet ‘in einem Punkte ’
as in our diagram and in Graper’s description. Instead, as
his figures show, the central meeting place is unstable territory. The group may resolve itself in three ways.13 One of
them is by retaining more or less of the contacts shown in
figure 9. A second is the arrangement in figure 11, which
would come from that in figure 10 if cell b formed contacts
with c and i, and cell i with b and h. Instead of a cluster of
six pentagons, there would then be two each of pentagons,
hexagons, and heptagons, with the average of six sides maintained. The third pattern arises when, in figure 10, cells a, h,
and i all acquire contacts with each other, becoming three
heptagons, and shutting out h, c, and e as three pentagons.
Graper does not analyze these groups even as far as we have
done, being content t o say of the arrangement in figure 8,“Allmahlich wird naturlich auch das ursprungliche Sechseckmosaik wiederhergestellt . . . .,, In the pigment layer
Wetzel found no rosettes or other unequivocal evidence of
Thompson: Growth and Form, p. 375.
141n a portion o f Wetzel’s drawing here reproduced as figure 12, there is a
group of six cells which we have marked with asterisks, exactly duplicating the
pattern in figure 11. There are two pentagons, two hexagons, and two heptagons,
all o f which may be made hexagonal by inserting a central hexagonal cell. We
do not infer, however, that a central cell has dropped out, since the same arrangement, as will be seen, can be explained on a different and more probable
Three radically tlift’cwiit I
of prodnciiig ail equal
number of pciitagons aiitl licptagoiis iii a hcsngonal epithclium liavc iiow heen consicleiwl, i i a m ~ ~ l y1)
, cell dii-isioii ; 2 )
iuiequal growth under special coiiclitioiis ; aiitl, 3 ) cell fusion
or elimination. Wctzcl, leaviiig out of consideratioil the first
alteriiati\.e, conltl not accouiit f o r the particular
of the various polygoiis. “1 t n.onlc1 certainly be ~ v o r t l i
wliilc,” he writes, “ t o subject t h c ~cwiiclit ions of tlic prodiictioii of iliese clift’erent
Fig 7
fig. 9
Accordingly we haye i.eproclncet1 ii portioii of his figure s h o w ing an appareiitly Iiapliaznrd arixiigemciit of s e ~ e i ipeutiigoiis, twelve liexagoiis, f o u r Iicptagoiis, m i d one octago1l
(fig.1 2 ) , and have aiialyzctl it in figure 13.
If it is assumed that i i i a group of eigliteeii r~g:.ul:~r
liexagoiis, six divide iii tlic 1)laiies intlicated iii figu1.e 1:3, they
will yield three liesagoiis aiid iiiiic peiittngoiis. Seren of the
latter bear the number 5 iii tlic figui*c;the other two we h a r e
made hesagoris 11)- division of tlic mai*ginal cells b a i i d fj.
Balaiiciiig the iiiiie pciitagoiis, seveii lieptagoiis aiicl o i i v
c ~ t a g o i iarc producctl by the divisions specified. F o u r of
these heptagons are not within the area. selected, being thc
margiiial cells u, c, tl, aiitl f . Air adclitioiial heptagon withiii
the group comes from the dix-isioii of the marginal cell 9. If
we re-draw figure 13 making c:very cell as nearly equilateral
;ISmay be, figure 14 is tlie result. It looks like epithelium, but
oiicl more in repose tliari that of figure 12, ~ ~ l i c r owing
somtl natural or artificial stress in the preparation, there is a
pronouiiced trend of the cells obliquely up and down the page.
1Cvery cell of the twentyfour lias hecn accounted for, both as
t o its num1)er of sidcs aiitl relative position.
An occasional area which caii be aiialyzecl in this way does
not prove the explanatioii correct. We know that a uniform
layer of hexagons cannot be tlie real starting point, but
ixtlier ;in area already including pentagons and heptagons
with perhaps further deviations. Flour of the six cell divisions in figure 13, whicli have been described as tlie division
of hexagons, can equally well be considered divisions of heptagons. Thns let c ~ l c,
l which is perliaps a heptagon, divide
first. It produces a lieptagon (beneath the octagon in the
figure) wliich divides next, and gives rise to a heptagon which
divides in turn (and so creates tlic octagon). It will readily
he fouiid that tlie entire set of divisions in figarc 2 can como
from the successive divisioit of heptagoiis. Since heptagons
arc' large cells, siich as would be expected to divide if a unif o r m size is to be maintained, the yucstioii arises wlietlier,
after tlie mosaic has become established, tlie division of lirlxagons is not an exceptional occurrence.
The bisection of a regular heptagon is accomplished by a
liricl passirig from the micldle of any side to the apex of the
angle opposite, ~7TTlierc*,with adjoining cells i t would make a
Ietralicdrnl angle. That would be avoitled by shiftiiig tlie
clivision-plane to oiie side or tlrc other, as ill figure 15 to ah.
i\ii unequal clirisioii of the lieptagon ensues, yielding a hexagon and a pentagon. In figure 15 tlie plane ab has been so
placrd that tlie resulting cells sliall have the proportionate
areas \ belong to regular hexagons and peiitagons re-
spectively; 60.2 per cent of the area goes to the former and
39.8 per cent to the latter. The pentagonal product, with an
area of 1.448, is 11.5 per cent larger than the pentagon derived from the bisection of the average hexagon. This considerable adrantage would facilitate its assimilation into the
Fig. 1 2 A n area of 2 1 ret,iiial pignient cells t,akcn from Wctzel’s “ R l ~ b . 1”
(Arrli. f. Entw-ineclianik, 1926, Bd. 107, p. 180) originally drawn expressly to
sliow the polygonal f o r m of the cells. Fig. 13, diagram showing how the
number of sicles and the relations t o each other of all the cells in figure 12 may b e
accounted for, through t,he division of regular hexagons. Fig. 14, diagram
constructed by attempting to ninkc the perimeter of every cell niarked o u t in
figure 13 niiniiiial f o r the area rnclost?d. I n all three drawings every polygon
other tlian a hexagon cmrluscs a fignrc indicating the numlit~rof its sidc3s.
mosaic. 'I!he hcxa.gozial product has still greater advantages ;
its area is 2.186, and by a growth of 18.8 per cent it becomes
R full-sized hexagonal ,cell. A pentagon derired from division
of an average hexagon must enlarge by 32.4 per cent., and
t,lien, after: receiving an acidit,ioiittl side from the c1ivisic)ii of
a n adjaceiit cell, must grow 50.1 per cent more, to accomplish
the same resnlt. Clearly, for inaint.aiziiiig tlic hexagona'l 1);i.t.t ~ r i itlie
, division of heptagons has decided advantages.
11 typical iastaiice of its occurrence is shown in figure 16.
Tlic t.woceiitra.1cells-a 1iexa.goiiand pentagon-have cert.aiiily
come from thc division of a common parent as indicat.ed by
the position of their nuclei arid t.lie thinness of t.he irit.ervening
wall. The parent cell was a heptagon, and its divisiori mtde
heptagons of the hexagons which were above and below it.
Thc cciitral peiitagoii anti hexagon arc now 42 and 58 per c.ent
i*espect.ively of their combined a.rea, as compared wit:ti the
geometric 39.8 and 60.2 per cent marked off in figure 1.5.
Should the upper 1i.ept.agoriin figure 16 now divide in thc
plane cxwssing the cell from the line a, t.he ent.ire group would
coiisist of t.hree Iiexagons, one pentagon, a~idone heptagon ;
h i t if the plane of divkion i i i the upper cell shoiild be directed
toward the licxagoii below, tho group would coiisist of one
I1ex2zg011,t.wo pentagons, aiid two 1icpt.agons. The prevalence
of 1iexagona.l cells l.liat tlie former altcmiative is
likely to be chosen, and surface tension offers, perhaps, ii
simple meeliartism for making the selection. For the pentagon, having a greater perimeter for its area than t.he hexagon,
may have a. greater tcridcricy t:o expand, a.iid so may muse
tlica c:mistr.ictioii tfuct to divisioii of tho adjaceiit cell to be
tlircctetl towaid it. rl'liat, Iiowctver, is a purely Iiypotlietical
ass ump tioii.
'l'he division of ail octagoii offers still greater atlva.iitagcs
t.lian t.liat of tlic heptagon. If of full size, it cmi divide int.o
It. is evident that a study of t.he incidenee of figurc-s in
smie suit,able siniple epithelium, for the purpose of finding in what shapes enil
s i m s of cells thcy oeeur most frequently, is a very promising program. Tlit? edls
must br oxamintd in surface view or horizontal sections. In stratifid cpitlitlliuni
R i tni1:tr olisnrvations would liw of significance u n l ( w the ent,irv cc+lls w w ~
a pair of hexagons which iieecl grow oiily 7.6 per cent t o acquire the average area of hexagons. Rut octagons and cells
with nine o r more sides are nsually not of full size. Formed
in tlie way that lias been describetl, it is not surprising that
often t,hey hare riot grown to tlic great extent necessary to
make their sides of unit length. The octagon in figure 12 is
no larger than an average heptagon, arid the dodecagon in
figlire 5 is uiiahle to fill out its territory and eliminate the
iiivadiiig mciiisci of tlie surrounding cells. The mechanical
Fig. 15
Fig. 16
Fig. 1.5 1111ision of a regular 1iclitag1111i n t o a n irregular hexagon and an
irregular pvnt:rgon, tlie awas of xhwh a r c t o eat11 otliei as t l i r arms of regulal
h c x ~ g o n t o rcgulnr pentagon of unit length of sirltl. Prom calculations 11%
Dr. H. W. Rrinkmann.
Fig. 16 A n instaricc of the d i
on of a Iic~ptagon into 11t~x;igonand pentagon
with t h r convcmion of two :uljoining hexagons into heptagons. Pignicnt Iapcr
of retina, hunian embryo of t5$ months. X 1000 diani. Note that division of
the upper hcptsgon i n the plane a xonld make tlic cntirc group c.onsist of tlilcc.
hcxngons, one p e n t a g o n , and one hrptagon.
readjustment wlieii sncli a cell divides is f ormidalole, and division is apt to stop with the nuclens. If coristrictioii in the
division-plane is prevciitctl, tlie 11ew cxll wall must, be more
than double the unit length, and siicli an extensive membrane
is riot formed. Thus the retiiial epithelium becomes dotted
with large many-sided hiniicleate cells. “111 tlie large cells
of tlie human retina, geiierally two riuclei may be recognized”
( TTTetzel). The small cells-quadrilaterals and all the pentamoiionucleate.
gons which I liappen to have observed-are
(iriiper considers that in ‘most organs’ there arc’ cells with
t \vo nuclei due to fusion. Otlicw regal-d birincleate cells as
oedcmatous or hypertrophic clegenerations, dividing throngli
amitosis. Rut if cell division is a method of maintainiiig uniform size in droplets of protoplasm, tlieri these Ininucleatch
cells are just such as should he in process of normal division.
‘L’licy presumably are in the telopliase of mitosis. Cptoplasmic division is retarded or prevented since the constriction
which it irivolves is mechanically difficnlt.
Thnis far, tlie pigmciit cells have heen coilsidered a s plane
siirfaccs. I n taking accwuiit of the liciglit of the cells, I have
tlependecl on calculations wliicli mere begun by Professor
IFT. C. Graustein; most of them, however, wire made during
his absence hy his assistant, Dr. H. W. Briilkmann, aiid the
following section of this report is largely theirs. Tlie height
of a prismatic cell of minimal surface is fomid by dividing
four times tlie area of the base by the perimeter,-
If’ tlrr l~asesa r e rcyqlar polygons with sides of unit) leiigtli
(\vhicli is made the unit> of measurement) their heights arc
t licreby fixed, and also their volumes ; and the possibility of
having a small surface for the volume decreases a s tlie sides
hecome fewer. In comparisoii with the surface of a sphere of
cqual volume ( C ) , their surfaces ( 8 ) meas1m as shown in
table 3 ; the quotient S / C may conveniently be called the surfmc. iiitlex of tlie cell or prism.
T o assist in visualizing these differences, projections of
the prisms of minimal surface have been drawn in figure 17.
Interest centers in tlic hexagoiial prism, which h a s the lowest
iiiclex of any form whicli can he joined to others of its kind
without interstices, anti consequently is the shape assumed by
cwlls in simple epithelium. A sigiiificaiit feature, not shown in
tfie projection, is the fact that the axes a, b , and c of figure I
exactly equal the height, in case the surface is minimal. The
name ‘cuboidal epithelium’ is justified, therefore, to the extcrrt that vertical sections of its cells in these plaiics sliould
he perfect squares ; aiid if the epithelium is neither stretched
nor compressed but rests a t ease, this condition is realized,
save that instead of being flat above and below, the cells
may be more or less rounded. TliornpsonlG has calculated
that tlie height of the rounded cap, in the case of cylindrical
drops or cells, should be 0.27 of the radius of the cylinder, and
that the entire disc would be that part of a spherical surface
subterided by a cone of 60”. Some rounding of the retinal
pigment cells has been generally recognized, and sometimes
the lateral walls arc not vertical but oblique,-which may be
important; hut in this further account it will he assumed that
the ends are flat aiitl the sides vertical.
I ,201
On examining vertical sections of the pigmented epithelium,
the impression is gained that the cells are not of the several
heights shown in figure 17, even if it is assumed that the cells
are so aligned as to divide the differences in elevation between
the top and bottom surfaces. There are slight irregularities,
but unfortunately there is no ready way to tell whether the
taller cells are many-sided aiid the shorter ones pentagons o r
quaclrilaterals. They all seem t o be very much of the same
height. Thompson says of them that they are “flattened to a
uniform thickness by the fluid pressure [of the vitreous body]
acting radially.” But in cuboidal epithelia subject to no such
pressure, it has not yet been shown that tlie cells vary in height
to accord with their several shapes in cross section. If the
pentagonal cells are not shorter than the minimal hexagons,
they have increased in volume against surface tension ; their
Growth and Form, page 2213.
iiiclclx, alrc~atlyhigh ( 1.201) hecomcs higher (1.208).
r 1
1 l i e Iieptagons m t l octagons, if rctlucetl to tlie liciglit of licsagoiis, also increasc their surface iiiclic~,-to 1.17(iO a r i d 1. l i M
~ ~ ~ s p e c t i v c ~This
l y . is apparently tlie actual coiiditicm in
c.nhoida1 epitlielia, tliougli a further study of the heights of
i1itlivichal cells is nectlctl. It means that the contacts of the
with foreign substances, at their f rec and basal siirfaces,
a y e made minimal. 111 stmtified epithelium, d i e r e tlic upper
siii*facc of a definite basal layer of cells is c o v e r ~ l1,)- cells
ctf similar substance, tlie upper surface of the basal layer is
110 longer flat. Its (~11s
map vary in heiglit, aiid arc facetetl
i i i m n t n d efforts toward minimal snrface. 1t woultl intc~rcst
El D
cytologist to kiiow the conformation of the upper and
nii(lc.r siurfaccs of mi oily emiilsioii, cwiitaining drops of TTari011s sizes, a s it floats 0x1 some watery fluid, but the physicists
c~oiisultctlas t o this 11i~vebeen somenkit cvasire. T'niform
hc~ipht,l i o \ v t ~ e r;itt:tincd, makes tlie volumes of the cells
tlirectly proportional t o their basal areas, ant1 so simplifies
tlw followiiig roliimeti~icstudy of their divisioii.
T h c ~hcsagoiial prism of minimal siirfaw is shoivii i n p v .jcc*tioii in figurt. 18,c'. Let it divide in halves along the plane
( I , 11, or c of figure 1 , anti the resiiltiiig cells ma>?retain their
oi.igiiiu1 liciglits by redwing their sides to 0.8fiN (fig. 18, B ) ;
or t Iiep may hecome rtlgular pc~iit:igoiialprisms of iinit sides
h? retluciiig thcir heiglits from 1.7321 to 1.2078 (fig. 18,B ) .
Rnt ctills of iicithcr of tliese types have hecii obsen-ccl. 111-
~ 1 ~ 7 1 OF
0 ~
stead, the irregular pentagons formed by such a division
seem to fill out to approximate regularity while retaining
their original heights, which means a growth in volume of
32.4 per cent. There is little doubt that much of this growth
is accomplished before division. Division of a heptagon of
average size produces a pentagon which need grow only 19
per cent to be of unit sides mid full height; and the largest
hexagons, which are the ones likeliest to divide, may be cut
into a pair of pentagons already full grown.
If the pentagonal cell is of the height of the hexagon of
minimal surface, it can reduce its surface index to the most
favorable figure possible for a pentagonal prism by increasing
the width of its sides to 1.26. This would raise the surface
Fig. 18 Projections to illustrate possible results of bisecting the hexagonal
prisni of minimal surfacc, C , having unit sides and a height of 1.7321. A and B
rrgular pentagonal prisms of half the voluinr of C: A, height 1.7321, sides 0.8691;
B, sides 1.0, height 1.30%.
and E, regular heptagonal prisms of the voluinr of
C: D, sides 1.0, height 1.2383; E, height 1.7321, sides 0.84556.
index of neighboring cells, however, and is perhaps never
fully realized. The largest pentagon of fourteen measured
had sides of 1.19; the smallest, of 0.89. I n part this range is
due to the varying size of the cells from which the pentagons
are cut off; in part, also, it represents growth-the large cells
being older than small ones-since there are no cells at hand
capable of producing the largest pentagons by simple division.
When thc hexagonal cell, figure 18, 6, becomes heptagonal
by the division of an adjacent cell, it may bccome regular and
retain its height by reducing its width of side to 0.8456 (fig.
18, E ) ; or it may have unit sides by reducing its height to
1.238 (fig. 18, D ) . Again, neither alternative is adopted, but
the cell grovvs so that it has the cross section of D and the
height of E , with surface index of 1.176. I have 110 eviclcncc
that it reclnces this index by increasing its lieight above the
general level axid acquiring the optimum proportions of fiq-
17, C.
rl’hc surface indices thus f a r recorded a r e those of geometrical Itrisms. A n approximate estimate of the indices f o r
tlic twenty-four cells included in figure 12 may be obtained
as follows. F r o m the area of each cell in surface view, its
hciglit may bc calculated, assuming that its surface is minimal. The avorage of these heights may tlieri be taken as the
iuniform thicliness of the epithelium. With the average leiigtli
of side of the cells in this group a s tlic unit of measarement.
t lie arerage of their heights proves to he 1.66-somewhat
shorter tlinii that of the regular hexagonal prism of minimal
surface, which is 1.73. After the volume of evei*y cell had
been determined on the arbitrary assumption of a uniform
height of 1.66, Doctor Brinkmann calculated their sui.face iiidices with the results shown in table 4.
For tlie tliicltriess post nlatccl, which is such as may actuall?
lie foiund uiider favorahlc conditions, tweiity-four cells of
tlic shapes outliiied in figure 13 have surface areas averaging
hut 1.5 per cent more tliari the theoretical minimal. Xiiice the
minimal prisms caniiot exist in combination witlioiit iiitersticcs, their cleviatioit from mathematical perfection may hc
distinctly less than 1.5 per cent. To this extent the reqniremciiits of minimal surface may be met, even though the heights
of the cells are made uniform.
A n inevitahlc question at this point is the natnrc of the
growth wliicli occurs when a cell receives ~ J Iadditional sicle
1)y division of its neighbor. There seems to be 110 evidence
cells. Conscthat it is at tlie expense of the surro~iiidii~g
qnently the growth mnst be due to the acquisition of subiiices from the tissue fluids beneath, and in this process
osmosis donlitless plays a large part. Wetzel, who supposes
that the turgor of the cells-their internal or osmotic pressure
--controls their shape eve11 in cross section, declares that
‘‘\vith equal turgor in all cells and in all directions, 0111)- ecyiiilire
lateral hexagonal prisms woulcl occur.” But in his figures, a s
in actnal preparations, the interfaces between pentagons, hexagons a i d heptagons a r e neither coiivex nor concave but conspicnously flat, compatible with an isotonic condition. There
a r e occasioiial exceptions, as in the dodecagon of figure 5
with its coilcave walls. Leduc in describing his artificial cells,
which a r e drops of a potassium ferrocyaiiide solution spreading over a gelatin plate, states explicitly that “the lines of
contact between the drops a r e straight when the solutions
a r e isotonic ; they a r e the more curved, the greater the difference in osmotic teiisioii of the liquids; the convexity is found
on the hvpertoiiic side, the concavity on the side of the h ~ p o -
i pc~itagoiis. . . . . . . . . . . . . ,
1 2 hexagons. . . . . . . . . . . . . .
4 hcptagoni: . . . . . . . . . . . . .
1 octagon . . . . . . . . . . . . . . .
1.211-1.24; A / . 1 . 2 %
1.19 1.21;
1 . 1 8 - 1 . 2 0 ; .I?. 1.190
1 .l(i.5
tonic liquid.’”7 The straightness or curvature of cell walls,
though sometimes significant i n this way, does not localize
growth in the cells which receive new sides, making it a n
osmotic phenomenon. It is a problem in that field in which
‘the whole future of cytology lies. ’
Time is sometimes considered a fourth dimension. The morphologist may deal with it by estimating the rate at which
new cells a r e added to the retinal epithelium. The posterior
half of the eyeball may, without great error, be considered
a perfect hemisphere. Its radius, i n a human embryo of 10
weeks (70 mm. C.-R.), wliich my assistant, Rlr. €3. J. Anson,
kindly dissected and measured f o r me, is 2.2 mm., and its
surface is therefore 30 Three moiiths later, in a n em-
’‘ Leduc, S.
p. 1301.
Diffusion tlans la gblatinc.
(”. lt., Acad. (1. Sci., Paris, 1901, T. 132,
206 mm. (C.-R.), A h . Xiisoii fiiids the radius 6.2 mm. :
the surface is tlieii 241 This eightfold iiicrease means
Illat cvery pigment wll, during the three months in qurstioii,
llns divided three times. Altlioiigli the ratti of oiie divisioii
ppr month f o r every cell is it great dccliiie from the origiiid
irnpcltus, it is still very high. Should it coiitiiiue unaBateil,
a t tlie age of five years the e ~ - e l ~~voulcl
exccetl the size of tlie
globc. At birth, from fig:uYes rccorclcd by JYeiss,ls
tllo r:idius of the pigrnented layer is pei* mm., aiicl the
i1rc1a of tlie posterior hemisphere is thcn 353 If the
(>ells1 l a 1 ~remained of the same size, 46 per cent of those
present a t *j1 months have tliricled oiice. After hirtli tlic
t ~ ~ - c ~ hgi*o\vs
rapidly diiriiig tlic first few )-c'ars, ant1 slo\vljtlici.raftei*, iuitil in the adult the area of this postcrior liemisplicrci is 830 sq.mrn. Assuming that cells a r e not lost ailti
I-eylncd-aiid therci seems t o he no c ~ k I e i i (that
~ tliep arc
riot as pcrmaiierit a s iierve cells,-then e r e i y pigment wll
tlivitles OIICP after I)ii.tli, hut only a small proportion of them
twice. 111this coiiclnsioii somc account is tak(w of theii. g ~ n ern1 increasci iii size.
At 5 Imoiitlis, the ttvcrage cliameter of fifty hexagonal ccfls;,
meas;uretl from m g l e t o angle, v a s foalid t o he 13.7 1-1, alicl the
i i w a of such ;I liexagon (corrected t o 96 per cent of the area of
a r*egular oiie) is 11'7 sq.1~. There arc, thrreforc, approsimately 2,050,000 pigment cells in the posterior half of thc
~j-cballat that stag:. if, a s stated in the preceding paragraph, 46 per cent 0-1' these cells divide diiriiig the 3; mouths
1)eforc birth, tlieii there is approximately m e cell tlivih i o i l per hour among 5500 cclls.
If the process of mitosis
takes two hours, a mitotic figure may lie expcctctl among
ovclry 2i50 cells. After hirtli they become much rarer, and
tliroug.11cytoplasmic growth the diameters of the pigment c ~ l l s
Imorne illcreased to 12--17 p according to Ronin,-i2-18 11,
TYhether the mitotic figiii~ls are miiformly distribiitcd
tliroi~gliout the Iiemispliere, or, as in so rnany organs, a i ~
1 ) 1 . ~ of
, 13. CIm d:rs SVnchstum des nrcnscli~ichcw~ u g r s . L h i : i t . H c f t v , .il)th.
I , 1x97, Utl. 8, 1'1'. 191-248.
limited to t.he periphery or other spccial regions, leaving large
groups of cells undist,urbed, requires further iiivestigati.on.
Apparently the gr0mt.h of thc subjaccnt layer of rods and
cones gently stretches the pigmciit layer in a.11 dircetions.
The pigment c.ells t,akc sustmancc from the uiiderlyiiig tissue
fluids and grow, a.nd t.hc largcst of t.hesc protoplasmic droplets here and there divide, forming pciitagons and heptagons
in the kaleidoscopic way which. this study may help to visualize. By the use of such rnet.liods as Voiiwiller has described,19
it may yet be possible, in some rapidly growiiig vegeta.hle
tissue, to observe this process in action.
uVonwiller, P. Xeue Wege der Gcwebelehrc, IT. Zeitschr. f . Anat. u. Entn.,
19.5, Bd. 7G, pp. 497-533.
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