<s>
Mathematical	B-Application
diagrams	I-Application
,	O
such	O
as	O
charts	O
and	O
graphs	B-Application
,	O
are	O
mainly	O
designed	O
to	O
convey	O
mathematical	O
relationships	O
—	O
for	O
example	O
,	O
comparisons	O
over	O
time	O
.	O
</s>
<s>
In	O
the	O
context	O
of	O
fast	O
Fourier	O
transform	O
algorithms	O
,	O
a	O
butterfly	B-Application
is	O
a	O
portion	O
of	O
the	O
computation	O
that	O
combines	O
the	O
results	O
of	O
smaller	O
discrete	B-Algorithm
Fourier	I-Algorithm
transforms	I-Algorithm
(	O
DFTs	O
)	O
into	O
a	O
larger	O
DFT	O
,	O
or	O
vice	O
versa	O
(	O
breaking	O
a	O
larger	O
DFT	O
up	O
into	O
subtransforms	O
)	O
.	O
</s>
<s>
The	O
name	O
"	O
butterfly	B-Application
"	O
comes	O
from	O
the	O
shape	O
of	O
the	O
data-flow	O
diagram	O
in	O
the	O
radix-2	O
case	O
,	O
as	O
described	O
below	O
.	O
</s>
<s>
The	O
same	O
structure	O
can	O
also	O
be	O
found	O
in	O
the	O
Viterbi	B-Algorithm
algorithm	I-Algorithm
,	O
used	O
for	O
finding	O
the	O
most	O
likely	O
sequence	O
of	O
hidden	O
states	O
.	O
</s>
<s>
The	O
butterfly	B-Application
diagram	I-Application
show	O
a	O
data-flow	O
diagram	O
connecting	O
the	O
inputs	O
x	O
(	O
left	O
)	O
to	O
the	O
outputs	O
y	O
that	O
depend	O
on	O
them	O
(	O
right	O
)	O
for	O
a	O
"	O
butterfly	B-Application
"	O
step	O
of	O
a	O
radix-2	O
Cooley	B-Algorithm
–	I-Algorithm
Tukey	I-Algorithm
FFT	I-Algorithm
algorithm	I-Algorithm
.	O
</s>
<s>
This	O
diagram	O
resembles	O
a	O
butterfly	B-Application
as	O
in	O
the	O
morpho	O
butterfly	B-Application
shown	O
for	O
comparison	O
,	O
hence	O
the	O
name	O
.	O
</s>
<s>
A	O
Venn	B-Application
diagram	I-Application
is	O
a	O
representation	O
of	O
mathematical	O
sets	O
:	O
a	O
mathematical	B-Application
diagram	I-Application
representing	O
sets	O
as	O
circles	O
,	O
with	O
their	O
relationships	O
to	O
each	O
other	O
expressed	O
through	O
their	O
overlapping	O
positions	O
,	O
so	O
that	O
all	O
possible	O
relationships	O
between	O
the	O
sets	O
are	O
shown	O
.	O
</s>
<s>
The	O
Venn	B-Application
diagram	I-Application
is	O
constructed	O
with	O
a	O
collection	O
of	O
simple	O
closed	O
curves	O
drawn	O
in	O
the	O
plane	O
.	O
</s>
<s>
A	O
Voronoi	B-Architecture
diagram	I-Architecture
is	O
a	O
special	O
kind	O
of	O
decomposition	O
of	O
a	O
metric	O
space	O
determined	O
by	O
distances	O
to	O
a	O
specified	O
discrete	O
set	O
of	O
objects	O
in	O
the	O
space	O
,	O
e.g.	O
,	O
by	O
a	O
discrete	O
set	O
of	O
points	O
.	O
</s>
<s>
This	O
diagram	O
is	O
named	O
after	O
Georgy	O
Voronoi	O
,	O
also	O
called	O
a	O
Voronoi	B-Architecture
tessellation	I-Architecture
,	O
a	O
Voronoi	B-Architecture
decomposition	I-Architecture
,	O
or	O
a	O
Dirichlet	B-Architecture
tessellation	I-Architecture
after	O
Peter	O
Gustav	O
Lejeune	O
Dirichlet	O
.	O
</s>
<s>
Each	O
site	O
s	O
has	O
a	O
Voronoi	B-Architecture
cell	I-Architecture
V(s )	O
consisting	O
of	O
all	O
points	O
closer	O
to	O
s	O
than	O
to	O
any	O
other	O
site	O
.	O
</s>
<s>
The	O
segments	O
of	O
the	O
Voronoi	B-Architecture
diagram	I-Architecture
are	O
all	O
the	O
points	O
in	O
the	O
plane	O
that	O
are	O
equidistant	O
to	O
two	O
sites	O
.	O
</s>
<s>
Wallpaper	O
groups	O
are	O
two-dimensional	O
symmetry	O
groups	O
,	O
intermediate	O
in	O
complexity	O
between	O
the	O
simpler	O
frieze	B-Algorithm
groups	I-Algorithm
and	O
the	O
three-dimensional	O
crystallographic	O
groups	O
,	O
also	O
called	O
space	O
groups	O
.	O
</s>
