# .

In mathematics, the Reeb foliation is a particular foliation of the 3-sphere, introduced by the French mathematician Georges Reeb (1920–1992).

It is based on dividing the sphere into two solid tori, along a 2-torus: see Clifford torus. Each of the solid tori is then foliated internally, in codimension 1, and the dividing torus surface forms one more leaf.

Reeb foliation, generated with gnuplot and POV-Ray

References

G. Reeb (1952). Sur certaines propriétés toplogiques des variétés feuillétées. Actualités Sci. Indust. 1183. Paris: Hermann.
Alberto Candel; Lawrence Conlon (2000). Foliations. American Mathematical Society. p. 93. ISBN 0-8218-0809-5.
Ieke Moerdijk; J. Mrcun (2003). Introduction to Foliations and Lie Groupoids. Cambridge studies in advanced mathematics 91. Cambridge University Press. p. 8. ISBN 0-521-83197-0.

Weisstein, Eric W., "Reeb Foliation", MathWorld.

---

set term povray
set output "~/prj/wikipedia/reebfoliation/torus.pov"
set view 40,280
set parametric
set iso 15
unset key
unset border
r=3
fx1(u,v)=(pi/2*sin(v)+r)*cos(u/(pi/2-0.3)*2*pi)
fy1(u,v)=(pi/2*sin(v)+r)*sin(u/(pi/2-0.3)*2*pi)
fz1(u,v)=pi/2*cos(v)
fx2(u,v,n)=(abs(u)*sin(v)+r)*cos(1/cos(u)+2*n*pi/6)
fy2(u,v,n)=(abs(u)*sin(v)+r)*sin(1/cos(u)+2*n*pi/6)
fz2(u,v,n)=abs(u)*cos(v)
splot [0:pi/2-0.3] [2*pi/4:6*pi/4] \
fx1(u,v), fy1(u,v), fz1(u,v) w l lt 6, \
fx1(u,v), fy1(u,v), fz1(u,v) w d lt 6,\
fx2(u,v,0), fy2(u,v,0), fz2(u,v,0) w l lt 2, \
fx2(u,v,0), fy2(u,v,0), fz2(u,v,0) w d lt 2,\
fx2(u,v,1), fy2(u,v,1), fz2(u,v,3) w l lt 3, \
fx2(u,v,1), fy2(u,v,1), fz2(u,v,3) w d lt 3,\
fx2(u,v,2), fy2(u,v,2), fz2(u,v,3) w l lt 2, \
fx2(u,v,2), fy2(u,v,2), fz2(u,v,3) w d lt 2,\
fx2(u,v,3), fy2(u,v,3), fz2(u,v,3) w l lt 3, \
fx2(u,v,3), fy2(u,v,3), fz2(u,v,3) w d lt 3, \
fx2(u,v,4), fy2(u,v,4), fz2(u,v,12) w l lt 2, \
fx2(u,v,4), fy2(u,v,4), fz2(u,v,12) w d lt 2, \
fx2(u,v,5), fy2(u,v,5), fz2(u,v,15) w l lt 3, \
fx2(u,v,5), fy2(u,v,5), fz2(u,v,3) w d lt 3,\
fx2(u,v,6), fy2(u,v,6), fz2(u,v,15) w l lt 2, \
fx2(u,v,6), fy2(u,v,6), fz2(u,v,3) w d lt 2,\
fx2(u,v,7), fy2(u,v,7), fz2(u,v,15) w l lt 3, \
fx2(u,v,7), fy2(u,v,7), fz2(u,v,3) w d lt 3

The following patch is then applied to pov source:

--- torus-orig.pov 2010-03-13 17:09:46.000000000 +0300
+++ torus.pov 2010-03-13 21:10:44.000000000 +0300
@@ -3,22 +3,22 @@
#include "colors.inc"
#include "shapes.inc"
#include "textures.inc"
-camera{ location <-291.511,-30.8094,-403.022>
+camera{ location <-291.511,-30.8094,-703.022>
sky <0,0,-1>
-look_at <23,25,-18>
-angle 13}
+look_at <23,25,-33>
+angle 5}
background{ color White }
-object{ Plane_XY texture{ PinkAlabaster }}
+//object{ Plane_XY texture{ pigment {White} }}
light_source{ <-291.511, -30.8094, -403.022> color 0.9* White }
light_source{ <-116.604,-12.3238,-282.116> color 0.9* White }
//color_definition
//#declare Pcolor0 = material{texture{pigment{color Red filter 0.6} finish{ambient 0.05 phong 0.8}}}
#declare Pcolor1 = material{texture{pigment{color Red}}}
-#declare Pcolor2 = material{texture{pigment{color Green}}}
-#declare Pcolor3 = material{texture{pigment{color Blue}}}
+#declare Pcolor2 = material{texture{pigment{color rgbt<0.5,0.9,0.5,0>}}}
+#declare Pcolor3 = material{texture{pigment{color rgbt<0.5,0.5,0.9,0>}}}
#declare Pcolor4 = material{texture{pigment{color Magenta}}}
#declare Pcolor5 = material{texture{pigment{color SkyBlue}}}
-#declare Pcolor6 = material{texture{pigment{color Brown}}}
+#declare Pcolor6 = material{texture{pigment{color rgbt<0.9,0.4,0.5,0>}}}
#declare Pcolor7 = material{texture{pigment{color Yellow}}}
#declare Pcolor8 = material{texture{pigment{color Orange}}}
#declare Grid = union{
@@ -65,9 +65,9 @@
cylinder{<0,25,0><50,25,0>,0.2}
cylinder{<0,37.5,0><50,37.5,0>,0.2}
}
-object{ Grid texture{DMFWood6}}
-text{ ttf "crystal.ttf","-5",0.2,0
-pigment{White}
+//object{ Grid texture{DMFWood6}}
+//text{ ttf "crystal.ttf","-5",0.2,0
+/*pigment{White}
scale 4
rotate<-40,0,280>
translate<3,-7,-1>
@@ -108,6 +108,7 @@
translate<-5,55,-37>
finish{ambient 3}
}
+*/
#declare Plot1 = union{
cylinder{<47.854,25,-36.6667>,<47.808,26.4495,-36.6667>,0.2}
cylinder{<47.808,26.4495,-36.6667>,<47.6701,27.8931,-36.6667>,0.2}
@@ -38492,4 +38493,3 @@
triangle{<22.5996,16.6939,-36.6667>,<21.481,17.0323,-36.6667>,<22.5554,16.5408,-31.9537>}
}
object{Plot18 material{Pcolor3}}