<s>
A	O
tetraview	B-Algorithm
is	O
an	O
attempt	O
to	O
graph	B-Application
a	O
complex	O
function	O
of	O
a	O
complex	O
variable	O
,	O
by	O
a	O
method	O
invented	O
by	O
Davide	O
P	O
.	O
Cervone	O
.	O
</s>
<s>
A	O
graph	B-Application
of	O
a	O
real	O
function	O
of	O
a	O
real	O
variable	O
is	O
the	O
set	O
of	O
ordered	O
pairs	O
(	O
x	O
,	O
y	O
)	O
such	O
that	O
y	O
=	O
f(x )	O
.	O
</s>
<s>
This	O
is	O
the	O
ordinary	O
two-dimensional	O
Cartesian	O
graph	B-Application
studied	O
in	O
school	O
algebra	O
.	O
</s>
<s>
A	O
tetraview	B-Algorithm
is	O
an	O
attempt	O
to	O
give	O
a	O
picture	O
of	O
a	O
four-dimensional	O
object	O
using	O
a	O
two-dimensional	O
representation	O
—	O
either	O
on	O
a	O
piece	O
of	O
paper	O
or	O
on	O
a	O
computer	O
screen	O
,	O
showing	O
a	O
still	O
picture	O
consisting	O
of	O
five	O
views	O
,	O
one	O
in	O
the	O
center	O
and	O
one	O
at	O
each	O
corner	O
.	O
</s>
<s>
A	O
picture	O
of	O
a	O
three-dimensional	O
object	O
is	O
a	O
projection	B-Algorithm
of	O
that	O
object	O
from	O
three	O
dimensions	O
into	O
two	O
dimensions	O
.	O
</s>
<s>
A	O
tetraview	B-Algorithm
is	O
set	O
of	O
five	O
projections	O
,	O
first	O
from	O
four	O
dimensions	O
into	O
three	O
dimensions	O
,	O
and	O
then	O
from	O
three	O
dimensions	O
into	O
two	O
dimensions	O
.	O
</s>
<s>
A	O
complex	O
function	O
w	O
=	O
f(z )	O
,	O
where	O
z	O
=	O
a	O
+	O
bi	O
and	O
w	O
=	O
c	O
+	O
di	O
are	O
complex	O
numbers	O
,	O
has	O
a	O
graph	B-Application
in	O
four-space	O
(	O
four	O
dimensional	O
space	O
)	O
R4	O
consisting	O
of	O
all	O
points	O
(	O
a	O
,	O
b	O
,	O
c	O
,	O
d	O
)	O
such	O
that	O
c	O
+	O
di	O
=	O
f( a	O
+	O
bi	O
)	O
.	O
</s>
<s>
To	O
construct	O
a	O
tetraview	B-Algorithm
,	O
we	O
begin	O
with	O
the	O
four	O
points	O
(	O
1	O
,	O
0	O
,	O
0	O
,	O
0	O
)	O
,	O
(	O
0	O
,	O
1	O
,	O
0	O
,	O
0	O
)	O
,	O
(	O
0	O
,	O
0	O
,	O
1	O
,	O
0	O
)	O
,	O
and	O
(	O
0	O
,	O
0	O
,	O
0	O
,	O
1	O
)	O
,	O
which	O
are	O
vertices	O
of	O
a	O
spherical	O
tetrahedron	O
on	O
the	O
unit	O
three-sphere	O
S3	O
in	O
R4	O
.	O
</s>
<s>
We	O
project	O
the	O
four-dimensional	O
graph	B-Application
onto	O
the	O
three-dimensional	O
sphere	O
along	O
one	O
of	O
the	O
four	O
coordinate	O
axes	O
,	O
and	O
then	O
give	O
a	O
two-dimensional	O
picture	O
of	O
the	O
resulting	O
three-dimensional	O
graph	B-Application
.	O
</s>
<s>
This	O
provides	O
the	O
four	O
corner	O
graph	B-Application
.	O
</s>
<s>
The	O
graph	B-Application
in	O
the	O
center	O
is	O
a	O
similar	O
picture	O
"	O
taken	O
"	O
from	O
the	O
point	O
of	O
view	O
of	O
the	O
origin	O
.	O
</s>
