<s>
Mathematical	O
phenomena	O
can	O
be	O
understood	O
and	O
explored	O
via	O
visualization	B-Application
.	O
</s>
<s>
Mathematical	B-Application
visualization	I-Application
is	O
used	O
throughout	O
mathematics	O
,	O
particularly	O
in	O
the	O
fields	O
of	O
geometry	O
and	O
analysis	O
.	O
</s>
<s>
Notable	O
examples	O
include	O
plane	O
curves	O
,	O
space	O
curves	O
,	O
polyhedra	O
,	O
ordinary	O
differential	O
equations	O
,	O
partial	O
differential	O
equations	O
(	O
particularly	O
numerical	O
solutions	O
,	O
as	O
in	O
fluid	O
dynamics	O
or	O
minimal	O
surfaces	O
such	O
as	O
soap	O
films	O
)	O
,	O
conformal	O
maps	O
,	O
fractals	B-Application
,	O
and	O
chaos	O
.	O
</s>
<s>
In	O
complex	O
analysis	O
,	O
functions	O
of	O
the	O
complex	O
plane	O
are	O
inherently	O
4-dimensional	O
,	O
but	O
there	O
is	O
no	O
natural	O
geometric	O
projection	O
into	O
lower	O
dimensional	O
visual	B-Application
representations	I-Application
.	O
</s>
<s>
Instead	O
,	O
colour	O
vision	O
is	O
exploited	O
to	O
capture	O
dimensional	O
information	O
using	O
techniques	O
such	O
as	O
domain	B-Application
coloring	I-Application
.	O
</s>
<s>
Visualization	B-Application
played	O
an	O
important	O
role	O
at	O
the	O
beginning	O
of	O
topological	O
knot	O
theory	O
,	O
when	O
polyhedral	O
decompositions	O
were	O
used	O
to	O
compute	O
the	O
homology	O
of	O
covering	O
spaces	O
of	O
knots	O
.	O
</s>
<s>
Permutation	O
groups	O
have	O
nice	O
visualizations	B-Application
of	O
their	O
elements	O
that	O
assist	O
in	O
explaining	O
their	O
structure	O
—	O
e.g.	O
,	O
the	O
rotated	O
and	O
flipped	O
regular	O
p-gons	O
that	O
comprise	O
the	O
dihedral	O
group	O
of	O
order	O
2p	O
.	O
</s>
<s>
The	O
cover	O
of	O
the	O
journal	O
The	O
Notices	O
of	O
the	O
American	O
Mathematical	O
Society	O
regularly	O
features	O
a	O
mathematical	B-Application
visualization	I-Application
.	O
</s>
