diff --git a/docs/BoundingBox.md b/docs/BoundingBox.md
index 275b81b..aa196e0 100644
--- a/docs/BoundingBox.md
+++ b/docs/BoundingBox.md
@@ -1,10 +1,10 @@
-## Bounding Boxes
+# Bounding Boxes
We want to investigate how bounding boxes behave in higher dimensions.
Bounding boxes are rectangular N-dimensional boxes encompassing an
ensemble of M (random) points, fullfilling some (likelihood) constraint.
-
+See our paper [Kester and Mueller (2021)](./references.md/#kester8).
We will plot 100 random points inside 4 N-balls of radius 1.0, for resp
N = [2,4,6,8], in black, red, green and blue. And their bounding boxes.
@@ -15,22 +15,21 @@ rejected. For an 8-ball only one in 73 points are OK. The random
points tend to concentrate to the middle.
-
| ndim | nsamples | rejected |
-|-:|-:|-:|
+|:-:|:-:|:-:|
| 2 | 100 | 22 |
| 4 | 100 | 233 |
| 6 | 100 | 991 |
| 8 | 100 | 7306 |
+
-This will not get us to a 1000-ball.
+This will not get us to a 1000-ball. We try something different.
-
-We try something else to get to a 1000-ball.
+## Distribution
We calculate the distribution of points thrown randomly into an N-ball,
as projected on a line through the center. It is obvious that this
@@ -43,11 +42,15 @@ For a 3-ball (cannonball) there is the volume of a circle present at
every x. It is proportional to
d3( x ) ~ ( 1 - x * x ) = d2( x ) 2
-For a 4-ball (hyperball) there the projection is a 3-ball, proportional to
+For a 4-ball (hyperball) there the projection is a 3-ball present at every x.
+It is proportional to
d4( x ) ~ d2( x ) 3
Etc.
+In the figure below we plotted the distribution for powers of, upto 1024.
+They are all scaled to a maximum of 1.
+
@@ -68,13 +71,15 @@ for an ensemble of M points will miss on average 1/M volume area.
N-balls (and other objects) in higher dimensions are quite
couterintuitive.
-Below we do some sanity checks, whether the distributions conform a
+## Sanity check
+
+Below we investigate, whether the distributions conform a
random ensemble of M=5000 points
-In the figure below, we have M/N points in N (=2,3,4,8,10) dimensions.
+In the figure below, we have 5000 points in N (=2,3,4,8,10) dimensions.
In green we see the calculated distribution scaled to a maximum of 1.0.
In red we have a histogram of the ensemble projected on each of the
-dimensional axes. M*5000 point in all, scaled to the same volume.
+dimensional axes. N*5000 point in all, scaled to the same volume.
On the right hand side we plot the moments of the distributions as
@@ -100,8 +105,10 @@ standard on uniformity. (made by rejection sampling).
The experiment follows the theory quite well.
+## More checks
+
Next we check the distribution of random points in 10 N-dim shells of
-equal volume and in 8 perpendicular sectors. We take 10000 points
+equal volume and in 8 perpendicular sectors. We take 10000 points,
random in spheres of 2,3,4,6, and 8 dimensions, by rejection sampling.
We expect 1000 points in each shell and 1250 in each sector.
@@ -115,8 +122,6 @@ The sectors are defined when ndim >= 3 and then dividing dimensions 1,
2, and 3 in its positive and negative values.
-
-
