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— 1
—
KEN
NET
H S
NEL
SON
, TH
E BI
NAR
Y W
ORL
D
byKe
nnet
h Sn
elso
n
NEW
TON
’S T
HIR
D L
AW A
ND
TH
E D
UA
LITY
OF
FORC
ES
TE
NS
EG
RI T
Y,
WE
AV
I NG
A
ND
TH
E B
I NA
RY
WO
RL
D
— 2
—
KEN
NET
H S
NEL
SON
, TH
E BI
NAR
Y W
ORL
D
Amon
g th
e te
rms f
or “t
won
ess”
are
dua
lity,
bin
ary,
twin
, pai
r, co
uple
, do
uble
, yin
-yan
g... T
hese
wor
ds re
fer t
o id
eas a
nd b
elie
fs in
art
, lite
ratu
re,
relig
ion,
scie
nce
and
philo
soph
y in
cul
ture
s eve
ryw
here
. Tw
ones
s wor
ds a
re
asso
ciat
ed w
ith g
ood-
evil,
mal
e-fe
mal
e, n
orth
-sou
th, p
ast-
futu
re, d
ay-n
ight
, up-
dow
n, p
ush
and
pull,
life
and
dea
th...
Stru
ctur
es o
f man
y ki
nds p
rovi
de e
vide
nce
of a
bin
ary
phen
omen
on
root
ed in
the
natu
re o
f thi
ngs,
with
exa
mpl
es in
the
cloc
kwise
-cou
nter
cloc
kwise
ro
tatio
n of
cog
-gea
rs, o
f com
pres
sion
vers
us te
nsio
n in
tens
egrit
y st
ruct
ures
, in
the
right
and
left
han
d he
lices
of f
abric
wea
ving
or s
impl
y in
the
reve
rsin
g of
co
lore
d sq
uare
s of a
che
ss b
oard
. Is
aac
New
ton’
s thi
rd la
w o
f mot
ion
stat
es c
lear
ly a
nd si
mpl
y th
e bi
narin
ess
in p
hysic
al fo
rces
: for
eve
ry a
ctio
n th
ere
is an
equ
al a
nd o
ppos
ite a
ctio
n. In
this
pict
ure
essa
y I d
escr
ibe
how
New
ton’
s thi
rd la
w a
nd d
ualit
y ar
e re
flect
ed in
m
any
kind
s of s
truc
ture
s.
THE
NA
TUR
E O
F ST
RU
CTU
RE
TE
NS
EG
RIT
Y,
WE
AV
ING
A
ND
TH
E B
INA
RY
WO
RL
D
— 3
—
KEN
NET
H S
NEL
SON
, TH
E BI
NAR
Y W
ORL
D
TH
E C
HE
CK
ER
BO
AR
D
FIG
. 1 FIG
. 3
In c
heck
erbo
ard
patt
erns
two
colo
rs a
ltern
ate
cell-
to-c
ell.
Chec
kerin
g is
a fig
ure-
grou
nd d
esig
n w
ith a
n ae
sthe
tic a
ll its
ow
n,
a vi
sual
syst
em fo
und
in th
e ar
t and
arc
hite
ctur
e in
cul
ture
s all
over
th
e w
orld
. W
heth
er c
ompo
sed
of p
olyg
ons (
Fig.
1) o
r ran
dom
shap
es
(Fig
. 2) t
he c
heck
erbo
ard
grid
disp
lays
the
prim
ary
beau
ty o
f bi
narin
ess w
here
nei
ghbo
rs a
re o
f opp
osite
col
or.
The
cros
sing
of tw
o lin
es, o
ne li
ne p
assin
g th
roug
h an
othe
r, (F
ig. 3
) is a
phe
nom
enon
, a fi
rst-
prin
cipl
es e
vent
that
initi
ates
a
chec
ker p
atte
rn. T
he in
ters
ectio
n w
here
the
lines
cro
ss d
ivid
es th
e pl
ane
into
qua
dran
ts w
hich
allo
w th
e bi
nary
col
orin
g of
alte
rnat
e ce
lls.
FIG
. 2
— 4
—
KEN
NET
H S
NEL
SON
, TH
E BI
NAR
Y W
ORL
D
A s
crib
ble
of c
lose
d lo
ops,
Fig.
4,
allo
ws
for t
wo-
colo
r che
cker
boar
ding
: Ea
ch lo
cus
whe
re li
nes
inte
rsec
t es
tabl
ishe
s fo
ur d
istin
ct a
reas
of
alte
rnat
ing
colo
rs.
In F
ig. 5
sm
all r
ed “b
ridge
s” a
re
supe
rimpo
sed
at e
ach
line-
cros
sing
s (m
agni
fied
in F
ig. 5
a) A
s in
wov
en fa
bric
th
e cr
ossi
ngs
alte
rnat
e ov
er-a
nd-u
nder
th
roug
hout
the
squi
ggle
figu
re. T
he
loop
ing
line
with
its
man
y cr
ossi
ngs
can
be v
iew
ed a
s a
wov
en k
not.
Thi
s be
autif
ul p
heno
men
on e
xem
plifi
es th
e es
sent
ial b
inar
ines
s of
nat
ure.
FI
G. 4
FIG
. 5FI
G. 5
A
BIN
AR
INE
SS
, A
NA
TU
RA
L P
HE
NO
ME
NO
N
CHEC
KERE
D S
CRIB
BLES
— 5
—
KEN
NET
H S
NEL
SON
, TH
E BI
NAR
Y W
ORL
D
BECA
USE
THE C
HEC
KERB
OAR
D PR
INCI
PLE C
ON
CERN
S AL
TERN
ATIN
G O
F NEI
GH
BORS
IT A
PPLI
ES A
LSO
TO TH
E BIN
ARY
CLO
CKW
ISE /
COU
NTE
RCLO
CKW
ISE R
OTA
TIO
N O
F GEA
R TRA
INS
(FIG
. 7) A
ND
THE N
ORT
H PO
LE/S
OU
TH PO
LE AT
TRAC
TIO
N O
F M
AGN
ETS W
ITH
POLA
RITY
ON
OPP
OSI
TE FA
CES:
NO
RTH
ON
ON
E FA
CE, S
OU
TH O
N TH
E OTH
ER. (
FIG
. 8 A
ND
8A)
FIG
. 7
FIG
. 8
FIG
. 8A
CL
OC
KW
ISE
/ C
OU
NT
ER
CL
OC
KW
ISE
BIN
AR
INE
SS
OF
GE
AR
S
DIS
K SH
APE M
AGN
ETS A
ND
CURR
ENT
LOO
P MAG
NET
S ATT
RACT
EDG
E-TO
-ED
GE W
HEN
POLE
S ARE
OPP
OSI
TE.
OR F
ACE-
TO-FA
CE
WH
EN N
ORT
H A
ND
SO
UTH
POLE
S ARE
IN
ALIG
NM
ENT
— 6
—
KEN
NET
H S
NEL
SON
, TH
E BI
NAR
Y W
ORL
D
ABO
UT
WEA
VIN
G
The
anci
ent i
nven
tion
of w
eavi
ng d
ispla
ys
the
basic
pro
pert
ies o
f nat
ural
stru
ctur
e: m
odul
ar-
repe
titio
n, le
ft a
nd ri
ght h
elic
al sy
mm
etry
and
the
clos
e as
soci
atio
n be
twee
n ge
omet
ry a
nd p
hysic
al
stru
ctur
e.Tw
o an
d on
ly tw
o fu
ndam
enta
l fab
ric w
eave
st
ruct
ures
exi
st: t
he st
anda
rd tw
o-w
ay p
lain
wea
ve
mad
e up
of s
quar
es, F
ig. 9
, and
the
thre
e-w
ay
tria
ngle
/hex
agon
wea
ve, F
ig. 1
0, u
sed
mos
t ofte
n in
ba
sket
ry. T
houg
h th
ere
are
man
y va
riatio
ns su
ch a
s cr
iss-c
ross
ing,
dou
blin
g, e
tc. t
hese
two
are
the
only
pr
imar
y fo
rms.
A sin
gle
wea
ving
eve
nt, t
wo
filam
ents
cro
ssin
g an
d in
con
tact
with
one
ano
ther
, Fig
. 11,
eac
h w
arpi
ng
the
othe
r whe
re th
ey p
ress
in c
onta
ct is
, in
itsel
f, an
el
emen
tary
stru
ctur
e. A
t the
poi
nt o
f cro
ssin
g th
e tw
o th
read
s cre
ate
dual
hel
ical
axe
s one
clo
ckw
ise, r
ight
-ro
tatin
g an
d th
e ot
her c
ount
ercl
ockw
ise, l
eft-
rota
ting.
Bypa
sses
, cro
sses
and
Xs h
ave
beco
me
pow
erfu
l sy
mbo
ls an
d sig
ns: a
chr
istia
n cr
oss,
skul
l and
cr
ossb
ones
, cro
ssed
fing
ers,
Xd
out,
sign
here
, kee
p ou
t. Th
is el
emen
tary
cro
ssin
g pr
inci
ple
refle
cts t
he
chec
kerb
oard
ing
scrib
ble’s
inte
rsec
tions
of F
ig. 5
. Thi
s bi
nary
pro
pert
y al
ong
with
the
right
and
left
han
d ro
tatio
n of
gea
rs F
ig. 7
, the
nor
th a
nd so
uth
pola
rity
of m
agne
ts F
ig. 8
, are
the
foun
datio
n of
bin
arin
ess.
Thes
e du
alite
s, th
ese
reve
rsal
s of r
ight
and
left
ro
tatio
n at
eve
ry c
ross
ing,
pro
vide
nat
ure’s
firs
t les
son
in fu
ndam
enta
l str
uctu
re. T
he h
elic
al p
heno
men
on
play
s a v
ital r
ole
in d
eter
min
ing
whe
ther
two
sepa
rate
pa
rts w
ill o
r will
not
link
toge
ther
.
WE
AV
ING
: M
OT
HE
R O
F T
EN
SE
GR
ITY
FIG
. 10
THRE
E-W
AY,
TRIA
NG
LE/H
EXAG
ON
WEA
VEIN
ASI
A: “K
AGO
ME”
FIG
. 9 T
HE C
OM
MO
N SQ
UARE
WEA
VE
COU
NTE
R-CL
OCK
WIS
EAX
IS
COU
NTE
R-CL
OCK
WIS
EAX
IS
CLO
CKW
ISE
AXIS
CLO
CKW
ISE
AXIS
FIG
. 11
Cros
s tw
o pe
ncils
. Pla
ce a
thum
b an
d in
dex
finge
r on
the
penc
ils a
nd sl
ide
tow
ard
the
cent
er.
Your
han
d w
ill te
nd to
rota
te e
ither
clo
ckw
ise o
r co
unte
rclo
ckw
ise.
— 7
—
KEN
NET
H S
NEL
SON
, TH
E BI
NAR
Y W
ORL
D RIG
HT
HE
LIX
, L
EF
T H
EL
IX
FIG
11.
KA
GO
ME
THRE
E-W
AY B
ASK
ET W
EAVE
In th
ree-
way
, or K
agom
e w
eavi
ng,
hexa
gons
alte
rnat
e w
ith tr
iang
les.
If th
e he
xago
ns h
ave
a cl
ockw
ise h
elix
the
tria
ngle
s ar
e co
unte
rclo
ckw
ise. I
f the
hex
agon
s are
co
unte
rclo
ckw
ise th
e tr
iang
les a
re c
lock
wise
.
FIG
. 10.
SQ
UARE
WEA
VEJu
st a
s the
indi
vidu
al c
ross
ings
of fi
lam
ents
hav
e al
tern
atin
g he
lical
axe
s so
each
squa
re in
a p
lain
wea
ve
alte
rnat
es w
ith it
s nei
ghbo
rs li
ke c
hess
boa
rd sq
uare
s. In
or
der t
o pr
ove
whe
ther
a w
eave
cel
l is r
ight
or l
eft h
ande
d,
imag
ine
your
fing
ers s
lidin
g in
con
tact
with
the
fram
e of
a c
ell. Y
our h
and
will
mov
e “do
wn-
hill”
in a
clo
ckw
ise/
coun
terc
lock
wise
sens
e ac
cord
ing
to th
e ce
lls “r
otat
ion”
.
— 8
—
KEN
NET
H S
NEL
SON
, TH
E BI
NAR
Y W
ORL
D
TH
E F
IVE
BA
SIC
WE
AV
E C
EL
LS
Belo
w a
re th
e fiv
e ba
sic w
eave
cel
ls. T
he fi
ve-
way
pen
tago
n is
used
onl
y in
bas
ket-w
eave
sphe
res
or c
ompo
und-
curv
atur
e ba
sket
s.
TWO
-WAY
CRO
SS U
NIT
THRE
E-W
AY TR
IIAN
GLE
UN
ITTW
O-W
AY P
LAN
E WEA
VE U
NIT
FIVE
-WAY
PEN
TAG
ON
WEA
VE U
NIT
REQ
UIR
ED IN
BA
SKET
-WEA
VE SP
HER
ESTH
REE-
WAY
HEX
AG
ON
WEA
VE U
NIT
RATT
AN
BA
SKET
-WEA
VE B
ALL
WIT
H O
UTL
INED
TRIA
NG
LE, P
ENTA
GO
N A
ND
HEX
AG
ON
CEL
LS.
THA
ILA
ND
CA
RVED
IVO
RY B
ALL
WIT
H
BASK
ET-W
EAVE
PATT
ERN
.C
HIN
A, 1
9TH
CEN
TURY
— 9
—
KEN
NET
H S
NEL
SON
, TH
E BI
NAR
Y W
ORL
D
E.Q
. CO
LUM
N, A
TEN
SEG
RITY
STRU
CTU
RE
SHO
WN
HER
E WIT
H O
VERL
AYED
BLU
E
CON
NEC
TIO
NS B
ETW
EEN
STRU
TS TO
IDEN
TIFY
THE W
EAVE
PATT
ERN
KELL
UM
’S G
RIP;
WO
VEN
WIR
E RO
PEW
OVE
N V
INYL
CO
LUM
N
The
thre
e co
lum
ns sh
own
here
shar
e an
iden
tity
with
bra
idin
g or
pl
aitin
g. T
he st
ruts
of t
he te
nseg
rity
colu
mn
(E. Q
. Tow
er) h
ave
a w
eave
pa
tter
n al
thou
gh th
ey a
re n
ot d
irect
ly c
onne
cted
to o
ne a
noth
er.
WO
VE
N C
OL
UM
NS
— 1
0 —
KEN
NET
H S
NEL
SON
, TH
E BI
NAR
Y W
ORL
D
The
x-m
odul
e co
lum
n al
so is
defi
ned
by
wea
ving
. On
the
left
is a
floa
ting
com
pres
sion
two-
way
or “
x-m
odul
e” c
olum
n sh
own
with
bl
ue c
onne
ctor
pat
hs su
perim
pose
d to
id
entif
y th
e co
mpr
essio
n m
embe
rs’ w
eave
pa
ths.
The
figur
e on
the
right
is a
vin
yl tu
bing
w
oven
col
umn
sect
ion
that
requ
ires f
our
inte
rwov
en tu
bes t
o em
ulat
e th
e te
nseg
rity
x-m
odul
e pa
tter
n. T
he e
xpre
ssio
n “t
wo-
way
” re
fers
to e
ach
x-m
odul
e’s se
t of t
wo
stru
ts.
WO
VE
N T
WO
-WA
Y,
X-M
OD
UL
E,
CO
LU
MN
X-M
OD
ULE
CO
LUM
NW
OVE
N V
INYL
-TU
BE C
OLU
MN
— 1
1 —
KEN
NET
H S
NEL
SON
, TH
E BI
NAR
Y W
ORL
D
The
wea
ve c
ells
show
n so
far r
elat
e to
po
lygo
ns; t
o tr
iang
les,
squa
res,
etc.
with
edg
es
that
byp
ass o
ne a
noth
er. I
t is p
ossib
le a
lso to
tr
ansla
te th
ree-
dim
ensio
nal s
olid
s, te
trah
edra
, oc
tahe
dra,
etc
., int
o w
eave
-like
cel
ls by
usin
g st
icks
as t
he p
olyh
edra
’s ed
ges.
I cal
l the
se
hybr
id c
onfig
urat
ions
“wea
ve-p
olyh
edra
”. Sho
wn
belo
w: a
wea
ve-te
trah
edro
n, a
wea
ve-t
runc
ated
-te
trah
edro
n, a
wea
ve-o
ctah
edro
n an
d a
wea
ve
cubo
ctah
edro
n. B
ecau
se o
f the
hel
ical
byp
ass a
t th
eir c
orne
rs th
ese
thre
e-di
men
siona
l str
uctu
res
have
all
the
char
acte
ristic
s of t
he fa
bric
wea
ve
cells
exc
ept t
hat e
ach
one
is a
spat
ial fi
gure
like
its
par
ent p
olyh
edro
n.
RE
GU
LA
R P
OL
YH
ED
RA
AN
DW
EA
VE
PO
LY
HE
DR
A
REG
ULA
R TE
TRA
HED
RON
WEA
VE TE
TRA
HED
RON
REG
ULA
R O
CTA
HED
RON
WEA
VE O
CTA
HED
RON
CU
BOC
TAH
EDRO
NW
EAVE
-CU
BOC
TAH
EDRO
N
TRU
NC
ATED
TETR
AH
EDRO
NW
EAVE
TRU
NC
ATED
TETR
AH
EDRO
N
— 1
2 —
KEN
NET
H S
NEL
SON
, TH
E BI
NAR
Y W
ORL
D
The
art o
f wea
ving
has
exi
sted
sinc
e th
e be
ginn
ing
of c
ivili
zatio
n. A
rche
olog
ists h
ave
turn
ed u
p ev
iden
ce o
f w
eavi
ng in
Egy
pt a
s ear
ly a
s 4,0
00 B
C. T
o ha
ve in
vent
ed th
is un
iver
sal c
raft
mus
t hav
e be
en in
deed
ast
onish
ing.
In 1
965
I was
exp
erim
entin
g w
ith m
odul
arly
repe
ated
te
nseg
rity
syst
ems w
hen
I beg
an to
und
erst
and
that
ther
e is
an u
nmist
akab
le fa
mily
-con
nect
ion
betw
een
tens
egrit
y an
d w
eavi
ng.
Abov
e is
a ph
oto
of m
e in
Sag
apon
ack,
New
York
, w
ith m
y fir
st d
owel
stic
k co
nstr
uctio
ns o
f tet
rahe
dral
(on
the
left
) and
oct
ahed
ral (
on th
e rig
ht) s
pace
fram
es. A
third
form
(in
the
cent
er) c
ompo
sed
of c
ubes
lack
s tria
ngul
atio
n w
hich
di
squa
lifies
it a
s a st
able
spac
e fra
me.
Whe
ther
my
disc
over
y w
as tr
uly
nove
l or m
erel
y a
redi
scov
ery
of so
met
hing
kno
wn
earli
er, p
erha
ps in
ano
ther
age
in a
noth
er c
ivili
zatio
n, is
im
poss
ible
to k
now
. The
U.S
. Pat
ent o
ffice
was
una
ble
to
turn
up
any
earli
er e
xam
ple.
#6,
739,
937
Thre
e-di
men
siona
l wea
ving
can
be
seen
as a
n ex
tens
ion
of c
onve
ntio
nal fl
at w
eavi
ng. I
n 3D
wea
ving
the
two-
way
and
thre
e-w
ay fl
at p
lane
s are
mad
e to
cris
s-cr
oss i
n or
derly
way
s tha
t giv
e ris
e to
“wea
ve-p
olyh
edra
” as
desc
ribed
on
the
follo
win
g pa
ges.
A lik
ely
reas
on sp
atia
l wea
ving
has
n’t b
een
disc
over
ed o
r pra
ctic
ed is
that
, whi
le th
ere
are
endl
ess u
ses
for f
abric
and
bas
ket w
eavi
ng, t
here
has
bee
n no
nec
essit
y fo
r spa
ce-fr
ame
wea
ving
.A
Kore
an e
ngin
eer,
Ki Ju
Kan
g, is
dev
elop
ing
an
appl
icat
ion
for 3
D sp
ace
fram
es. H
e ha
s bee
n ex
perim
entin
g w
ith th
ree-
dim
ensio
nal w
eavi
ng h
e ha
s nam
ed, “
Wire
-w
oven
bul
k Ka
gom
e” tr
usse
s. Ki
Ju h
opes
his
plan
ar w
ire
trus
ses w
ill b
e ad
apte
d in
man
ufac
turin
g hi
gh-s
tren
gth
stee
l sa
ndw
ich-
pane
ls fo
r shi
p bu
ildin
g an
d ai
rcra
ft te
chno
logy
.
WO
VE
N 3
-DIM
EN
SIO
NA
LS
PA
CE
-FR
AM
ES
KI JU
KAN
G, W
IRE B
ULK
KAG
OM
E SPA
CE FR
AME M
OD
EL,
7 X 7
X 2.
5 IN
. GIF
T TO
SN
ELSO
N, 2
012.
— 1
3 —
KEN
NET
H S
NEL
SON
, TH
E BI
NAR
Y W
ORL
D
Ratt
an o
ctah
edro
n/cu
boct
ahed
ron
wov
en sp
acef
ram
e
Four
wea
ve-o
ctah
edra
, Illu
stra
ting
the
basic
thre
e-di
men
siona
l oct
ahed
ron/
cubo
ctah
edro
n w
eave
pa
tter
n. In
the
cent
er o
f the
gro
up is
a sq
uare
that
id
entifi
es th
e cu
boct
ahed
ron;
onl
y ha
lf co
mpl
ete
in
this
four
-cel
l exa
mpl
e.
OC
TAH
ED
RO
N/C
UB
OC
TAH
ED
RO
NW
OV
EN
SP
AC
EF
RA
ME
WEA
VE O
CTA
HED
RON
WEA
VE C
UBO
CTA
HED
RON
— 1
4 —
KEN
NET
H S
NEL
SON
, TH
E BI
NAR
Y W
ORL
D
Five
wea
ve-te
trah
edra
, sho
win
g th
e ba
sic th
ree-
dim
ensio
nal t
etra
hedr
al w
eave
pat
tern
com
pose
d of
wea
ve-te
trah
edra
and
wea
ve-t
runc
ated
-te
trah
edra
alte
rnat
ing
in sp
ace
with
one
ano
ther
.
This
spat
ial w
eave
pat
tern
has
four
diff
eren
t dire
ctio
ns
of tr
iang
le/h
exag
on fl
at w
eave
pla
nes.
Thes
e re
peat
ed p
lane
s al
ign
with
the
face
pla
nes o
f a n
orm
al te
trah
edro
n. T
he a
ltern
ate,
la
rger
form
s, th
e w
eave
-tru
ncat
ed-te
trah
edra
, occ
upy
the
cavi
ties i
n be
twee
n th
e w
eave
-tetr
ahed
ra.
Ratt
an te
trah
edro
n/tr
unca
ted-
tetr
ahed
ron
wov
en sp
acef
ram
e.
TE
TR
AH
ED
RO
NS
PA
CE
FR
AM
E W
EA
VE
WEA
VE T
ETRA
HED
RON
WEA
VE T
RUN
CAT
ED
TETR
AH
EDRO
N
— 1
5 —
KEN
NET
H S
NEL
SON
, TH
E BI
NAR
Y W
ORL
D
FR
OM
WE
AV
ING
TO
TE
NS
EG
RIT
Y
— 1
6 —
KEN
NET
H S
NEL
SON
, TH
E BI
NAR
Y W
ORL
D
WE
AV
E C
EL
LS
IN
TO
TE
NS
EG
RIT
Y C
EL
LS
X-M
OD
ULE
; CO
MPL
ETE T
RIA
NG
ULA
TIO
N3-
WAY
PRI
SM;
COM
PLET
E TRI
AN
GU
LATI
ON
SQU
ARE
PRI
SM; S
QU
ARE
S A
RE N
ON
-TRI
AN
GU
LATE
DPE
NTA
GO
NA
L PRI
SM; P
ENTA
GO
NS
ARE
NO
N-T
RIA
NG
ULA
TED
HEX
AG
ON
AL P
RISM
; HEX
AG
ON
S A
RE N
ON
-TRI
AN
GU
LATE
D
Wea
ving
and
tens
egrit
y sh
are
the
prin
cipl
e of
alte
rnat
ing
helic
al
dire
ctio
ns, o
f lef
t-to
-rig
ht, o
f byp
asse
s cl
ockw
ise a
nd c
ount
ercl
ockw
ise.
In th
e to
p ro
w a
bove
are
fiv
e pr
imar
y w
eave
figu
res.
Belo
w
them
are
the
equi
vale
nt te
nseg
rity
mod
ules
. Ind
ivid
ual t
ensio
n lin
es --
st
rings
, wire
s or r
ope
-- ar
e at
tach
ed
to th
e en
ds o
f the
stru
ts a
s sho
wn
so
that
eac
h as
sem
bly
is a
clos
ed sy
stem
m
ade
of te
nsio
n an
d co
mpr
essio
n pa
rts.
Each
tens
ion
line
conn
ects
in
divi
dual
ly to
the
ends
of t
wo
stru
ts.
They
do
not t
hrea
d th
roug
h lik
e a
strin
g of
bea
ds. T
he te
nsio
n lin
es
mus
t be
adju
sted
for t
ight
ness
as
with
tuni
ng a
strin
ged
inst
rum
ent o
r
infla
ting
a ca
r tire
.Ti
ghte
ning
the
tens
ion
syst
em
stor
es b
oth
tens
ion
and
com
pres
sion
forc
es in
equ
al a
mou
nts,
a st
ate
that
en
gine
ers c
all “
pres
tres
sing.
” Th
e en
ergy
rem
ains
stor
ed in
side
the
stru
ctur
e un
til it
is d
isass
embl
ed.
In th
e fig
ures
abo
ve, o
nly
the
2-st
rut “
x-m
odul
e” a
nd th
e 3-
stru
t pr
ism h
ave
tens
ion
netw
orks
with
to
tal t
riang
ulat
ion.
The
net
wor
ks
of th
e sq
uare
pris
m, t
he p
enta
gon
prism
and
the
hexa
gon
prism
are
not
co
mpo
sed
of tr
iang
les.
In te
nseg
rity
stru
ctur
es tr
iang
ulat
ion
in th
e te
nsio
n ne
twor
k is
signi
fican
t bec
ause
it
dete
rmin
es if
the
stru
ctur
e w
ill b
e fir
m o
r not
.
Tens
egrit
y st
ruct
ures
are
en
dosk
elet
al, a
s are
hum
ans a
nd
othe
r mam
mal
s who
se te
nsio
n “m
uscl
es” a
re e
xter
nal t
o th
e co
mpr
essio
n m
embe
rs’ b
ones
.U
niqu
e to
tens
egrit
y, th
e co
mpr
essio
n st
ruts
are
sepa
rate
d on
e fro
m a
noth
er, n
on-to
uchi
ng w
ithin
th
eir t
ensio
n en
velo
pe. T
he e
xcep
tion
is th
e tw
o-st
rut x
-mod
ule,
or
trad
ition
al k
ite fr
ame.
Thi
s ess
entia
lly
flat fi
gure
’ lack
s a c
ompr
essio
n fo
rce
in th
e “z”
dire
ctio
n. In
ord
er to
se
para
te th
e cr
osse
d st
ruts
a th
ird
stru
t or e
lse
an a
dditi
onal
X-m
odul
e,
mus
t be
adde
d to
pul
l the
two
stru
ts
apar
t.
— 1
7 —
KEN
NET
H S
NEL
SON
, TH
E BI
NAR
Y W
ORL
D WE
AV
E P
OL
YH
ED
RA
AN
D
TE
NS
EG
RIT
Y P
OL
YH
ED
RA
TEN
SEG
RITY
TETR
AH
EDRO
N
TOTA
L TRI
AN
GU
LATI
ON
TEN
SEG
RITY
OC
TAH
EDRO
N;
TRIA
NG
ULA
TED
NET
WO
RK EX
CEP
T
FOR
ITS S
IX SQ
UA
RE FA
CES
WEA
VE O
CTAH
EDRO
N; S
AME A
S ABO
VE:
THE V
ERTI
CES (
RED
) HEL
IXES
ARE C
LOCK
-W
ISE.
THE F
ACES
(BLU
E) AR
E CO
UN
TER-
CLO
CKW
ISE
WEA
VE T
ETRA
HED
RON
; TH
E VE
RTIC
ES
(RED
) H
ELIX
ES A
RE C
LOCK
WIS
E. TH
E FA
CES (
BLU
E) AR
E CO
UN
TERC
LOCK
WIS
E
— 1
8 —
KEN
NET
H S
NEL
SON
, TH
E BI
NAR
Y W
ORL
D
A rig
ht-h
ande
d te
nseg
rity
mod
ule
can
be tr
ansf
orm
ed in
to a
left-
hand
ed o
ne b
ut it
mus
t be
com
plet
ely
reco
nstr
ucte
d, e
xcha
ngin
g al
l par
ts
in re
latio
n to
one
ano
ther
star
ting
with
the
twist
rela
tions
hip
of th
e co
mpr
essio
n st
ruts
. The
righ
t and
left
confi
gura
tions
are
mirr
or im
ages
of o
ne
anot
her.
Wha
t are
the
cons
eque
nces
of
this
reve
rsib
ility
? It i
s tha
t in
the
mirr
ored
figu
res,
the
dire
ctio
nal s
ense
of
all
pres
tres
sed
pull-
and-
push
forc
es
are
also
reve
rsed
. The
tens
ion
forc
es
that
pul
l cou
nter
-clo
ckw
ise in
a le
ft-ha
nded
form
pul
l clo
ckw
ise in
a ri
ght-
hand
ed fo
rm a
nd v
ice
vers
a.
In c
olum
n-st
ruct
ures
ther
e is
an a
dvan
tage
to a
ltern
atin
g he
lical
di
rect
ions
mod
ule-
to-m
odul
e be
caus
e th
e in
here
nt fl
exib
ility
of a
tens
egrit
y
stru
ctur
e is
in it
self
helic
al. W
hen
pres
sed
on, a
righ
t-ha
nded
mod
ule
rota
tes s
light
ly to
the
left
and
vic
e ve
rsa,
so th
at th
e en
tire
tow
er st
ruct
ure
flexe
s whe
n co
mpr
esse
d to
p-to
-bo
ttom
. By
alte
rnat
ing
mod
ules
righ
t-le
ft-rig
ht-le
ft h
elic
al ro
tatio
n in
the
colu
mn
com
es to
zer
o.
RIG
HT
HE
LIX
/ L
EF
T H
EL
IXC
ON
NE
CT
ING
MO
DU
LE
S T
OG
ET
HE
R
TOP
VIEW
TOP
VIEW
TOP
VIEW
SID
E VIE
W
SID
E VIE
W
SID
E VIE
W
TOP
MO
DU
LERI
GH
T HEL
IX
MID
DLE
MO
DU
LELE
FT H
ELIX
BOTT
OM
MO
DU
LERI
GH
T HEL
IX
— 1
9 —
KEN
NET
H S
NEL
SON
, TH
E BI
NAR
Y W
ORL
D
TR
IAN
GU
LA
TE
D T
EN
SIO
N N
ET
WO
RK
SIt
is po
ssib
le to
con
stru
ct a
ny n
umbe
r of v
arie
d te
nseg
rity
confi
gura
tions
, fro
m s
impl
e to
hig
hly
com
plex
. Yet
, onl
y th
ose
form
s w
hose
tens
ion
netw
ork
is co
mpo
sed
entir
ely
of t
riang
les
are
trul
y st
able
. If
the
netw
ork
has
squa
res,
pent
agon
s, et
c. t
he s
truc
ture
will
lac
k fir
mne
ss. T
his
is es
peci
ally
true
of t
ense
grity
sph
eres
, non
e of
whi
ch h
ave
tria
ngul
ated
tens
ion
netw
orks
.
Type
1:T
ENSI
ON
/CO
MPR
ESSI
ON
TRI
ANG
LE.
Wor
king
muc
h lik
e a
sling
use
d by
rigg
ers
for h
oist
ing,
tria
ngle
s of t
ype
1 ar
e fo
rmed
w
ith tw
o st
ruts
and
two
tend
ons.
The
two
tens
ion
lines
run
from
the
end
of o
ne
stru
t to
the
two
ends
of a
seco
nd st
rut.
Type
2: T
ENSI
ON
-ON
LY T
RIAN
GLE
.
A te
nseg
rity
tria
ngle
can
als
o be
form
ed w
ith th
ree
tens
ion
lines
att
ache
d to
thre
e di
ffere
nt st
ruts
.
A th
ree-
stru
t pris
m sh
owin
g ty
pe 1
, red
and
type
2 g
reen
tr
iang
les
The
two
diffe
rent
tria
ngle
type
s ide
ntifi
ed
on a
pho
to o
f Nee
dle
Tow
er a
t the
Hirs
hhor
n M
useu
m a
nd S
culp
ture
Gar
den
in
Was
hing
ton,
D.C
.
The
tria
ngle
s in
a te
nseg
rity
netw
ork
are
form
ed in
two
diffe
rent
way
s, de
signa
ted
as ty
pe 1
and
type
2 tr
iang
les.
— 2
0 —
KEN
NET
H S
NEL
SON
, TH
E BI
NAR
Y W
ORL
D
TE
NS
ION
TR
IAN
GL
ES
CO
NT
INU
ED
Beca
use
all t
ensi
on li
nes,
strin
gs, w
ires,
cabl
es, h
ave
som
e de
gree
of
elas
tic s
tret
ch,
tens
egrit
y st
ruct
ures
the
mse
lves
ar
e el
astic
and
spr
ingy
dep
endi
ng o
n th
e tig
htne
ss o
f th
e pr
estr
essi
ng a
nd th
e ch
arac
teris
tics
of th
e te
nsio
n m
ater
ial.
The
elas
tic fl
exin
g of
a t
ense
grit
y st
ruct
ure,
a c
olum
n fo
r exa
mpl
e, c
an b
e se
en in
the
smal
l rot
atio
ns o
f the
righ
t or
left
hel
ixes
. A ri
ght-
hand
ed h
elix
com
pres
ses
with
left
rota
tion
and
vice
ver
sa.
The
tow
er s
culp
ture
sho
wn
here
is
as a
n ex
celle
nt
exam
ple.
All
tens
ion
lines
-- e
dges
, slin
gs, d
raw
s --
are
of e
qual
le
ngth
so
that
all
type
2 tr
iang
les,
colo
red
gree
n in
the
pict
ure,
ar
e eq
uila
tera
l. W
hen
pres
sed
dow
n on
, and
then
rele
ased
, the
co
lum
n re
spon
ds li
ke a
coi
led
sprin
g.
Its
nam
e is
Equ
ilate
ral
Qui
verin
g To
wer
.
MO
DEL
OF E
QUI
LATE
RAL Q
UIVE
RING
TO
WER
— 2
1 —
KEN
NET
H S
NEL
SON
, TH
E BI
NAR
Y W
ORL
D
FOLD
LIN
ES O
F TY
PE 1
TRI
AN
GLE
S.
Type
1 tr
iang
les o
ccur
alw
ays i
n pa
irs li
ke b
utte
rfly
win
gs.. T
he fo
ld
line
of e
ach
win
g-pa
ir of
tria
ngle
s is
simila
r to
a cr
ease
in th
e fo
lded
-pap
er
colu
mn
to th
e rig
ht. T
he sp
ringi
ness
of
tens
egrit
y st
ruct
ures
hap
pens
in
the
hing
ing
alon
g th
e “fo
ld li
nes”
.
Kite
fram
e st
ruct
ure
with
its
two
stru
ts a
nd fo
ur te
nsio
n lin
es is
co
mpo
sed
only
of t
ype
1 tr
iang
les.
Type
1 tr
iang
les o
ccur
her
e al
so
in p
airs
mak
ing
a di
amon
d fo
rm.
In th
e ki
te fr
ame,
two
face
-to-
face
dia
mon
ds sh
are
the
tens
ion
lines
on
oppo
site
face
s of t
he k
ite.
Not
e th
at th
is m
ost e
cono
mic
al o
f st
ruct
ures
is a
ctua
lly a
flat
tene
d te
trah
edro
n.
This
fold
ed p
aper
col
umn
simul
ates
the
geom
etry
of a
thre
e-w
ay
tens
egrit
y co
lum
n. T
he ty
pe 1
tria
ngle
s ar
e re
d an
d th
e ty
pe 2
are
in g
reen
. U
nlik
e te
nseg
rity,
this
pape
r fac
simile
is
not a
pre
stre
ssed
stru
ctur
e. S
till,
it se
ems l
ikel
y th
at th
e va
lley
and
hill
fold
s ev
en o
f ran
dom
ly c
rinkl
ed p
aper
bea
r a
clos
e re
latio
nshi
p to
tens
egrit
y’s t
ensio
n an
d co
mpr
essio
n pa
tter
ns o
f for
ce
TE
NS
ION
TR
IAN
GL
ES
CO
NT
INU
ED
— 2
2 —
KEN
NET
H S
NEL
SON
, TH
E BI
NAR
Y W
ORL
D
TH
RE
E T
YP
ES
OF
TE
NS
ION
LIN
ES
A te
nseg
rity
stru
ctur
e’s w
hole
tens
ion
netw
ork
is ex
tern
al to
the
com
pres
sion
stru
ts so
that
it is
an
endo
skel
etal
st
ruct
ure
with
com
pres
sion
forc
es
push
ing
out a
gain
st th
e te
nsio
n sk
in.
Each
sepa
rate
line
con
nect
s tw
o po
ints
ed
ges,
draw
s and
slin
gs a
nd e
ach
type
pl
ays a
spec
ific
role
in th
e te
nsio
n ne
twor
k.
Edge
tens
ion
lines
defi
ne th
e ed
ges a
nd th
e sid
es
of e
ach
mod
ule.
The
thre
e-w
ay c
olum
n ha
s thr
ee
edge
lin
es fo
r eac
h m
odul
e. In
mos
t cas
es, e
dges
ca
rry
less
tens
ion
than
dra
w o
r slin
g lin
es.
Dra
w te
nsio
n lin
es p
ull t
he m
odul
es to
war
d on
e an
othe
r. In
the
thre
e-w
ay c
olum
n, e
ach
mod
ule
is co
nnec
ted
to e
ach
neig
hbor
by
thre
e as
cend
ing
draw
s and
thre
e de
scen
ding
dra
w li
nes.
Slin
g te
nsio
n lin
es su
spen
d th
e m
odul
es,
perf
orm
ing
like
the
sling
s use
d in
rigg
ing
wor
k. T
hey
conn
ect o
ne m
odul
e to
the
next
and
are
gen
eral
ly in
opp
ositi
on to
the
draw
line
s. In
a th
ree-
way
col
umn
six sl
ings
ar
e re
quire
d in
ord
er to
link
two
mod
ules
.
To th
e rig
ht is
a th
ree-
way
mod
ule.
In th
is sim
ple
figur
e al
l ten
sion
lines
can
be
calle
d ed
ges:
six e
nd-e
dges
and
thre
e sid
e-ed
ges.
They
defi
ne, r
ough
ly, a
tria
ngul
ar p
rism
. Whe
n vi
ewed
thro
ugh
the
vert
ical
axi
s the
mod
ule
has
a le
ft-ro
tatio
n he
lix. T
he o
ppos
ite is
true
whe
n vi
ewed
from
the
side:
the
stru
ts re
late
to o
ne
anot
her i
n a
cloc
kwise
or r
ight
-rota
tion
helix
.
RIG
HT H
ELIX
LEFT
HEL
IX
RIG
HT H
ELIX Th
ree
mod
ule,
thre
e-w
ay c
olum
n
— 2
3 —
KEN
NET
H S
NEL
SON
, TH
E BI
NAR
Y W
ORL
D
The
simpl
e ki
te fr
ame,
two
cros
sed
stru
ts h
eld
firm
ly
toge
ther
by
a gi
rth
of fo
ur te
nsio
n m
embe
rs, i
s a h
uman
in
vent
ion
and
prob
ably
thou
sand
s of y
ears
old
. Lon
g be
fore
pe
ople
cov
ered
it w
ith p
aper
for u
se a
s a fl
ying
obj
ect t
o lo
ft in
th
e w
ind
the
fram
e m
ost l
ikel
y w
as u
sed
as a
ligh
twei
ght p
alle
t, a
stre
tche
r for
tran
spor
ting
thin
gs. B
asic
as i
t is,
the
pres
tres
sed
kite
exi
ts o
nly
in th
e w
orld
of p
eopl
e; n
ot a
s a p
rodu
ct o
f nat
ure.
Th
e ki
te fr
ame
can
be b
uilt
in m
any
prop
ortio
ns a
s sho
wn
belo
w. T
he st
ruct
ural
prin
cipl
e re
mai
ns th
e sa
me
exce
pt th
at
the
dist
ribut
ion
of fo
rces
, bot
h te
nsio
n an
d co
mpr
essio
n, v
ary
as th
e pr
opor
tions
are
alte
red.
Alw
ays,
thou
gh, t
he to
tal o
f the
co
mpr
essio
n fo
rces
pus
hing
out
are
equ
al to
the
sum
of t
he
tens
ion
forc
es p
ullin
g in
.Th
e ki
te fr
ame
is qu
asi-t
ense
grity
bec
ause
the
two
stru
ts,
lack
ing
a fo
rce
in th
e “z”
dire
ctio
n in
ord
er to
sepa
rate
, tou
ch
and
pres
s on
one
anot
her w
here
they
cro
ss. T
he k
ite st
ruct
ure
is th
e ba
sic p
rest
ress
ed te
nsio
n-co
mpr
essio
n ce
ll fo
r x-m
odul
e te
nseg
rity
stru
ctur
es.
The
leng
ths o
f the
four
tend
ons a
nd th
e le
ngth
s of t
he
stru
ts d
eter
min
e th
e sh
ape.
The
kite
fram
e’s te
nsio
n an
d co
mpr
essio
n sy
stem
. The
stru
ts
push
out
(com
pres
sion)
and
the
tend
ons p
ull i
n (te
nsio
n).
TH
E K
ITE
-FR
AM
E X
-FO
RM
T
EN
SE
GR
ITY
’S P
RIM
AR
Y
Kite
fram
e sh
ape
varia
tions
— 2
4 —
KEN
NET
H S
NEL
SON
, TH
E BI
NAR
Y W
ORL
D TH
E K
ITE
-FR
AM
E B
EC
OM
ES
TH
RE
E D
IME
NS
ION
AL
The
kite
fram
e is
tran
sfor
med
into
a
true
tens
egrit
y st
ruct
ure
with
a th
ird
stru
t whi
ch is
add
ed b
y re
plac
ing
one
of
the
orig
inal
edg
e te
nsio
n lin
es (i
n gr
een)
w
ith fo
ur n
ew li
nes s
how
n in
red.
The
se
four
per
form
as s
lings
that
susp
end
the
new
stru
t. Th
e th
ree-
stru
t str
uctu
re m
ust
now
be
mad
e st
able
by
addi
ng tw
o ad
ditio
nal l
ines
- dr
aws-
- tho
se sh
own
in b
lue.
The
dra
w te
ndon
s go
from
the
ends
of t
he n
ew, t
hird
, str
ut to
the
far
ends
of t
he o
rigin
al p
air;
to th
ose
ends
th
at w
ill d
raw
the
kite
stru
ts a
way
from
on
e an
othe
r. Co
nnec
ted
to th
e w
rong
tw
o en
ds th
e dr
aw li
nes w
ill o
nly
forc
e th
e ki
te st
ruts
into
firm
er c
onta
ct a
nd
fail
to a
chie
ve a
floa
ting
com
pres
sion
stru
ctur
e.
It is
esse
ntia
l in
this
simpl
e st
ruct
ure,
as i
n al
l ten
segr
ity st
ruct
ures
, to
est
ablis
h th
e op
timum
leng
ths f
or
the
tens
ion
mem
bers
so th
at th
e w
ork
will
be
firm
and
tigh
tly p
rest
ress
ed.
This
can
only
be
done
by
succ
essiv
e ad
just
men
ts, b
y tr
ial a
nd e
rror
. If t
he
leng
th o
f one
line
is c
hang
ed th
e te
nsio
n on
all
lines
are
effe
cted
.
As a
gen
eral
rule
the
draw
line
s ar
e th
e pr
imar
y m
eans
for i
ncre
asin
g or
re
laxi
ng th
e pr
estr
essin
g of
a te
nseg
rity
stru
ctur
e. A
s with
mos
t rul
es, t
here
are
a
varie
ty o
f exc
eptio
ns.
This
cons
truc
tion
proc
ess,
addi
ng
part
s one
by
one,
has
now
tran
sfor
med
th
e ba
sic k
ite fr
ame
into
a p
rope
r th
ree-
stru
t ten
segr
ity m
odul
e.. I
t is
stru
ctur
ally
the
sam
e as
the
thre
e-w
ay
mod
ule
show
n on
pag
e 15
. Onl
y th
eir
shap
es a
nd sy
mm
etrie
s are
alte
red
by
size
adju
stm
ents
of t
he te
ndon
s and
st
ruts
.
Repl
acin
g A
Sin
gle
Stru
tW
ith A
n X-
Mod
ule
In th
e fig
ure
to th
e rig
ht, t
he th
ird st
rut
that
was
intr
oduc
ed in
the
figur
e ab
ove
is its
elf r
epla
ced
by a
seco
nd k
ite fr
ame;
sli
ngs i
n re
d, d
raw
s in
blue
. Thi
s new
as
sem
bly
of tw
o “x
-mod
ules
” rep
rese
nts
the
first
step
in a
con
stru
ctio
n pr
oces
s --
addi
ng m
odul
e af
ter m
odul
e --
that
can
be
exp
ande
d in
defin
itely
. Eac
h op
en
quad
rant
of a
ny m
odul
e off
ers a
pla
ce
to c
onne
ct y
et a
noth
er x
-mod
ule.
A ki
te-fr
ame
tran
sfor
med
in
to a
tens
egrit
y st
ruct
ure
A tw
o-m
odul
e X-
colu
mn
— 2
5 —
KEN
NET
H S
NEL
SON
, TH
E BI
NAR
Y W
ORL
D
X-M
OD
UL
E E
XP
AN
DE
D
The
term
spac
e-fil
ling
appl
ies t
o su
ch sy
stem
s, fo
r ex
ampl
e, a
s sug
ar c
ubes
pac
ked
in a
box
or o
rang
es a
t th
e m
arke
t. Th
e x-
mod
ule
too
can
be e
xpan
ded,
add
ing
mod
ule
afte
r mod
ule,
in a
ll di
rect
ions
. Bel
ow a
re e
xam
ples
of
how
“X” m
odul
es c
an b
e re
peat
ed in
defin
itely
.
Dou
ble
Star
195
0-20
02
X-m
odul
e 90
deg
ree
corn
er is
cre
ated
by
addi
ng a
third
mod
ule.
An e
xpan
ded
X-m
odul
e pl
ane
— 2
6 —
KEN
NET
H S
NEL
SON
, TH
E BI
NAR
Y W
ORL
D TE
NS
EG
RIT
Y A
DA
PT
S T
OA
VA
RIE
TY
OF
FO
RM
S
Osa
ka
Six #
2
Six
#1No
rthw
ood
III
Ladl
e Pi
ece
El C
ordo
bes
Thes
e sc
ulpt
ures
are
com
pose
d of
six
stru
ts
— 2
7 —
KEN
NET
H S
NEL
SON
, TH
E BI
NAR
Y W
ORL
D
TH
E B
INA
RY
GE
OM
ET
RY
OF
MA
GN
ET
S
— 2
8 —
KEN
NET
H S
NEL
SON
, TH
E BI
NAR
Y W
ORL
D
A un
ique
gro
up o
f five
pol
yhed
ra h
ave
a sp
ecia
l qua
lity:
they
per
mit
the
chec
kerin
g of
adj
acen
t fac
es. T
his n
atur
al b
inar
y pr
oper
ty
mak
es it
pos
sible
to c
onst
ruct
them
usin
g po
lygo
n-sh
aped
refri
gera
tor m
agne
ts w
hose
no
rth
and
sout
h po
les a
re o
n op
posit
e fa
ces l
ike
head
s and
tails
of a
coi
n. W
hen
asse
mbl
ed th
e m
agne
ts sn
ap to
geth
er e
dge-
to-e
dge
to c
reat
e a
firm
mag
net p
olyh
edro
n.
BIN
AR
Y P
OL
YH
ED
RA
AN
D M
AG
NE
TS
Oct
ahed
ron
8 tr
iang
le m
agne
tsCu
boct
ahed
ron
14 m
agne
ts
Icos
idod
ecah
edro
n32
mag
nets
Smal
l Rho
mbi
cubo
ctah
edro
n26
mag
nets
Rhom
bico
sidod
ecah
edro
n62
mag
nets
— 2
9 —
KEN
NET
H S
NEL
SON
, TH
E BI
NAR
Y W
ORL
D
Eigh
t-m
agne
t oct
ahed
ra a
nd fo
urte
en-
mag
net c
uboc
tahe
dra
can
be a
ssem
bled
toge
ther
to
form
an
expa
ndab
le sp
ace-
fillin
g m
atrix
with
bi
nary
mag
netic
bon
ding
from
cel
l to
cell.
MA
GN
ET
IC A
RC
HIT
EC
TU
RE
— 3
0 —
KEN
NET
H S
NEL
SON
, TH
E BI
NAR
Y W
ORL
D 8 G
EARS
10 G
EARS
5 G
EARS
18 G
EARS
32 G
EARS
14 G
EARS
5 M
AG
NET
S8
MA
GN
ETS
10 M
AG
NET
S14
MA
GN
ETS
18 M
AG
NET
S32
MA
GN
ETS
SP
HE
RIC
AL
GE
AR
TR
AIN
S,
MA
GN
ET
IC A
ND
ME
CH
AN
ICA
LPi
ctur
ed b
elow
are
two
sets
of g
ear t
rain
s. In
the
uppe
r ro
w th
e ge
ars a
re m
ade
of d
isc-s
hape
mag
nets
with
nor
th a
nd
sout
h po
les o
n op
posit
e fa
ces l
ike
the
mag
net-
poly
hedr
a on
the
prev
ious
two
page
s. Th
ese
sphe
rical
bin
ary
mos
aics
hav
e th
e pe
culia
r num
ber:
5, 8
, 10,
14,
18
and
32.
The
disc
-sha
pe m
agne
ts in
the
top
row
are
cer
amic
m
agne
ts su
ppor
ted
on n
on-m
agne
tic a
rmat
ures
. The
y sn
ap
toge
ther
edg
e-to
-edg
e th
roug
h no
rth-
sout
h at
trac
tion
in th
e sa
me
man
ner a
s the
pol
yhed
ron
mag
nets
but
bec
ause
they
are
roun
d an
d in
mag
netic
con
tact
they
can
revo
lve
as se
ts o
f gea
rs.
If on
e m
agne
t is m
ade
to re
volv
e by
han
d th
e ot
hers
follo
w a
s ge
ars.
In th
e bo
ttom
row
are
com
pute
r im
ages
of m
etal
gea
rs
that
hav
e th
e sa
me
num
bers
and
geo
met
ry a
s the
mag
net
sphe
res o
f the
top
row
.M
echa
nica
l or m
agne
tic, a
ll of
thes
e bi
nary
phe
nom
ena
are
first
prin
cipl
es ru
les o
f fun
dam
enta
l geo
met
ry; l
aws t
hat
dete
rmin
e th
e de
sign
of st
ruct
ures
in n
atur
e.
Computer images, Jon Monaghan
— 3
1 —
KEN
NET
H S
NEL
SON
, TH
E BI
NAR
Y W
ORL
D
Cloc
kwise
and
cou
nter
cloc
kwise
rota
tion,
a b
inar
y pr
inci
ple
as w
ell a
s a sy
mm
etry
prin
cipl
e, c
an b
e se
en in
m
any
way
s in
vario
us st
ruct
ures
. Whe
n I u
se th
e w
ord
“rot
atin
g” to
des
crib
e th
e or
der o
f thr
eads
in a
wea
ve
patt
ern
or th
e ar
rang
emen
t of s
trut
s in
a te
nseg
rity
stru
ctur
e it
is cl
ear t
hey
are
in fa
ct si
ttin
g st
ill. I
nsid
e of
th
e st
ruct
ure
thou
gh, t
he fo
rces
are
act
ing
in e
ither
a
cloc
kwise
or c
ount
ercl
ockw
ise d
irect
ion.
Eve
n so
, thi
s he
lical
tend
ency
is tr
ansla
tabl
e in
to a
ctua
l mot
ion
by
tran
spos
ing
the
stat
ic d
omai
ns o
f ten
segr
ity fi
gure
s int
o ac
tual
whe
els o
r gea
rs. S
how
n he
re a
re th
ree
tens
egrit
y st
ruct
ure
exam
ples
alo
ng w
ith se
ts o
f disk
-sha
ped
mag
nets
. The
se sp
heric
al a
rran
gem
ents
are
bor
n of
the
sam
e ge
omet
ry a
s the
tens
ion
netw
orks
of t
he te
nseg
rity
stru
ctur
es.
As w
ith th
e po
lyhe
dral
mag
net m
osai
cs th
e di
sc
mag
nets
show
n he
re h
ave
thei
r nor
th p
oles
on
one
face
an
d so
uth
on th
e ot
her s
o th
at w
hen
they
are
edg
e-to
-ed
ge, c
heck
erbo
arde
d w
ith o
ppos
ite p
oles
faci
ng o
ut,
they
snap
toge
ther
. Rot
atin
g on
e m
agne
t by
hand
cau
ses
the
othe
rs fo
llow
as a
gea
r-tra
in.
The
mag
net s
truc
ture
s are
a p
art o
f my
mul
timed
ia,
artw
ork,
“Por
trai
t of a
n At
om”. T
hey
also
tell
us th
at
the
wor
ld o
f str
uctu
re a
nd g
eom
etry
is a
hal
l of m
any
mirr
ors e
ndle
ssly
refle
ctin
g sim
ilarit
ies,
rela
tions
hips
an
d nu
mbe
rs. I
n th
e ca
se o
f the
mag
nets
, the
dire
ctio
ns
the
arro
ws p
oint
des
crib
e no
t onl
y th
e ro
tatio
n an
d co
unte
r-rot
atio
n of
the
gear
s but
they
also
iden
tify
the
dire
ctio
n el
ectr
ons w
ould
be
mov
ing
if th
ese
wer
e cu
rren
t el
ectr
ical
loop
s rat
her t
han
perm
anen
t mag
nets
, in
orde
r to
pro
duce
nor
th/s
outh
mag
netic
att
ract
ion.
Tho
ugh
my
atom
mod
el, “
Port
rait
of a
n At
om” i
s spe
cula
tive,
the
mag
net r
elat
ions
hips
and
thei
r geo
met
ry a
re a
fact
of
natu
re.
A tw
elve
-str
ut te
nseg
rity
stru
ctur
e sh
own
side-
by-s
ide
of it
s co
unte
rpar
t, a
sphe
rical
four
teen
-mag
net s
et. t
he si
tes f
or th
e m
agne
ts a
re id
entifi
ed w
ith th
e tw
elve
-str
ut fo
rm’s
eigh
t tria
ngul
ar
corn
er tr
iang
les a
nd si
x sq
uare
face
s.
Six-
stru
t ten
segr
ity st
ruct
ure
and
an e
ight
-mag
net s
pher
ical
gea
r set
. Th
e co
rner
tria
ngle
s of t
he si
x-st
rut fi
gure
are
alte
rnat
ely
cloc
kwise
and
co
unte
rclo
ckw
ise h
elix
es. T
he e
ight
-whe
el sp
heric
al se
t also
has
alte
rnat
ing
mag
nets
that
are
che
cker
boar
ded
arou
nd th
e sp
here
.
Thre
e-st
rut t
ense
grity
and
five
-mag
net s
pher
ical
gea
r set
. The
hel
ix
on th
e to
p an
d bo
ttom
tria
ngle
s are
cou
nter
cloc
kwise
; the
edg
es a
re
cloc
kwise
. The
se c
orre
spon
d to
the
alte
rnat
ing
of m
agne
ts a
nd w
heel
ro
tatio
n in
the
sphe
rical
set o
f five
mag
nets
.
BIN
AR
Y P
RIN
CIP
LE
SS
TAT
IC A
ND
KIN
ET
IC
— 3
2 —
KEN
NET
H S
NEL
SON
, TH
E BI
NAR
Y W
ORL
D
EX
TE
ND
ED
MA
GN
ET
IC
GE
AR
TR
AIN
S
A bo
dy-c
ente
red
cubi
c ar
rang
emen
t of
14-m
agne
t sph
eres
. Eac
h sp
here
con
nect
s to
its n
eigh
bors
at t
he c
orne
r pos
ition
s of a
cu
be.
Thes
e m
agne
t ass
embl
ies,
like
the
indi
vidu
al m
agne
t sph
eres
, are
uni
t gea
r tra
ins:
one
of th
e m
agne
ts is
mad
e to
turn
by
hand
th
e re
st w
ill fo
llow
in u
niso
n. I
disc
over
ed th
e
mag
net s
truc
ture
s ove
r fift
y ye
ars a
go, i
n 19
62.
From
that
disc
over
y w
as b
orn
my
fasc
inat
ion
with
the
atom
’s el
ectr
onic
arc
hite
ctur
e.
A bo
dy-c
ente
red
arra
y of
8-m
agne
t sph
eres
. If
this
patt
ern
is ex
tend
ed in
defin
itely
eac
h 8-
mag
net u
nit w
ill h
ave
8 ne
ighb
ors a
t its
co
rner
pos
ition
s.
— 3
3 —
KEN
NET
H S
NEL
SON
, TH
E BI
NAR
Y W
ORL
D
5-m
agne
t cel
ls in
a h
exag
on b
eehi
ve p
atte
rn. M
agne
tic
linka
ge is
con
tinuo
us. I
n m
y at
om m
odel
, thi
s hex
agon
fo
rmat
ion
repr
esen
ts th
e ar
rang
emen
t of c
arbo
n at
oms
in a
pla
ne o
f gra
phen
e.
A cu
bic
form
com
pose
d of
8-
mag
net s
pher
es a
ltern
atin
g w
ith 1
4-m
agne
t sph
eres
in
perf
ect m
agne
tic c
ontin
uity
. Th
e po
larit
ies o
f the
adj
acen
t ce
lls h
ave
reve
rse
pola
rity.
If
a 14
-mag
net s
pher
e ha
s
its 8
cor
ner-m
agne
t sou
th
pole
s fac
ing
out a
nd it
s 6
face
-mag
net’s
nor
th p
oles
fa
cing
out
, its
nei
ghbo
ring
14-m
agne
t set
will
orie
nt
thes
e po
larit
ies i
n re
vers
e.
— 3
4 —
KEN
NET
H S
NEL
SON
, TH
E BI
NAR
Y W
ORL
D
Thes
e pr
inci
ples
of s
truc
ture
are
the
foun
datio
n of
my
mod
el o
f the
ato
m. S
ee th
ese
inte
rest
ing
pape
rs:
atom
file
s at k
enne
thsn
elso
n.ne
t/th
e-at
omht
tp://
kenn
eths
nelso
n.ne
t/Po
rtra
itOfA
nAto
m.p
dfht
tp://
kenn
eths
nelso
n.ne
t/Sn
elso
nAnA
rtist
sAto
m.p
dfht
tp://
kenn
eths
nelso
n.ne
t/Sn
elso
n_Ci
rcle
sSph
eres
AndA
tom
s.pdf
http
://ke
nnet
hsne
lson.
net/
artic
les/
KSne
lson_
Pape
r_FQ
Xi_u
pdat
ed.p
dfht
tp://
kenn
eths
nelso
n.ne
t/ar
ticle
s/In
dust
rialD
esig
nFeb
1963
