This paper establishes a bijection between the number of regions formed when a circle is cut by chords and the number of regions formed when 4-dimensional hyperspace is cut by hyperplanes. It first introduces counting methods to determine the number of regions in each case using formulas. It then presents labeling algorithms to map each region in the circle cuts to a unique region in the hyperspace cuts, proving their counts are equal. Further work is needed to fully label the circle cut regions to match the labeling of the hyperspace cuts.

1. A BIJECTION BETWEEN HYPERSPACES AND PLANES
AUCKLY, DAVID & SAYYARI, MOHAMMED
ABSTRACT
Finding the number of regions when space is cut
by a number of planes is a well known problem
in combinatorics. It may be generalized to vari-
ous dimensions. A different problem is to com-
pute the number of regions when a circle is cut
by chords. One interesting fact we noticed is that
the number of regions of a circle cut by chords is
exactly the same as the number of regions when
4-dimensional space is cut by hyperplanes.
METHOD
The problem is split into two stages. The counting
stage, and the correspondence stage.
Counting. Find formulas for counting the circle
cut by chords. Then, the 4D Hyperspace cut by
hyperplanes. Establish a relationship between the
discovered phenomenons.
Correspondence. Find a common labeling of the
4D hyperspaces and the planes to establish a bi-
jective correspondence.
LABELING ALGORITHMS
Labeling Cuts Algorithm 1 (LCA1)
1. Cut with a new line c.
2. Insert from the left a "+" for the old regions.
3. Insert from the left a "-" for the new regions.
-+
++ -+
--new
old
Figure 6: A region cut once is cut a second time.
Labeling Cuts Algorithm 2 (LCA2)
1. Cut the region with the cut c.
2. Increment the size of the labeling bits by 1.
3. Assign the region to the left of the cut with a "-"
at the bit c and "+"’s everywhere else.
4. Assign "+" at the index c for all the other re-
gions.
(-++)
(+++)
(+-+)
(++-)
Cut 1
Cut 2
Cut c
Figure 7: A region cut twice is cut a third time.
Algorithm To Determine New Regions
1. Add the new dot d at the 0th
dot.
2. Slide the new dot towards the d − 1st
dot.
3. The denote the emerging regions as new.
0
4
1 2
3
0
1
2
3
4
0
1 2
3
4
Stage 1 Stage 2 Stage 3
Figure 8: Finding which regions to label as new.
COUNTING ALGORITHMS
Circle Cut by Chords
Pd = 1 +
d
2
+
d
4
(1)
From now on we will call it regions.
Figure 3: Circles cut by Chords (Weisstein)
3
2
1
0
5
4
Figure 4: Circle cut by chords of 6 dots.
Circle Cut by Lines
rn
c =
n
i=0
c − 1
i
(2)
Figure 5: Circles cut by Lines (Weisstein)
REFERENCES
Weisstein, Eric W. "Circle Division by Chords." From MathWorld–A Wolfram Web Resource.
http://mathworld.wolfram.com/CircleDivisionbyChords.html
Weisstein, Eric W. "Circle Division by Lines." From MathWorld–A Wolfram Web Resource.
http://mathworld.wolfram.com/CircleDivisionbyLines.html
WORK IN PROGRESS
Labeling the circle cut by chords
Find an algorithm for labeling the circles cut by
dots such that the labeling pattern matches that
of the hyperspaces cut by hyperplanes.
CONTACT INFORMATION
Web 7matto.me
Email M.k.Alsayyari@gmail.com
Email mohammed.alsayyari@kaust.edu.sa
Phone +966 (53) 635 9933
Phone +1 (720) 325 0999
INTRODUCTION
First, we must introduce some counting methods and definitions.
Counting By Choosing
5Choose2 or
5
2
represents the number of ways of choosing two items out of five. For example if the
five items are a, b, c, d, e there are 10 ways to choose 2, namely, ab, ac, ad, ae, bc, bd, be, cd, ce, de.
Bijection
Domain Range
Figure 1: A Bijection.
A Bijection is a
correspondance
taking every element
in the domain, to
exactly one element
in the range and
vice-versa.
Illegal Configurations
Any intersection has a maximum of two crossing
lines.
Illegal
Legal
Figure 2: Legal and illegal configurations
Counting by Adding
Let n be the dimension our regions live in, c the
number of cuts, and rn
c the number of regions. Let
d be the number of dots on the outside of a circle
and Pd be the number of parts the circle is cut into
after joining the dots with chords.
Dot d Pd r4
c r3
c r2
c r1
c r0
c Cut c
3 4 4 4 4 3 1 2
4 8 8 8 7 4 1 3
5 16 16 15 11 5 1 4
Table 1: Results of counting regions and planes.