Minimal cluster computation for four planar regions with the same area
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Citations
The quadruple planar bubble enclosing equal areas is symmetric
On the Steiner property for planar minimizing clusters. The anisotropic case
On the Steiner property for planar minimizing clusters. The anisotropic case
References
The standard double bubble is the unique stable double bubble in
Planar Soap Bubbles
Minimal clusters of four planar regions with the same area
Planar Soap Bubbles
Related Papers (5)
Frequently Asked Questions (14)
Q2. What is the simplest way to obtain a stationary cluster?
if a stationarycluster is obtained as the stereographic projection of a partition on the sphere, the authors can rotate the sphere aswe like, project back the partition to the plane and obtain another stationary cluster.
Q3. How do the authors get the triple bubbles?
All triple bubbles (i.e. the minimal clusterswith three regions, see [15]) are obtained, up to rescaling, by projecting the geodesic network of a regulartetrahedron on the sphere.
Q4. How many arcs are there in R2?
Wichiramala [14] proved that for N = 3 in R2 the three regions of any minimizer are delimited by six circular arcs joining in four points.
Q5. What is the simplest way to identify the ambient plane?
Let us identify the ambient plane with the set of complex numbers C. Up to translating, rotating and rescalingwemight suppose that the two vertices of the resulting double bubble are the two complex numbers 0 and 1.
Q6. What is the topology of the cluster?
It is conjectured that in this topology there is a unique minimizer which is the cluster with two axes of symmetry: regions E 1 and E 2 are congruent to each other and the same is true for regions E 3 and E 4 .
Q7. What is the simplest way to check stationarity?
One can check (see [13],[5], [15], [11]) that stationarity is also preserved by circle inversion: if three circular arcs meet in a point withequal angles of 120 degrees of course their circular inversion are also arcs (or straight line segments) whichalso meet in a point with the angles preserved.
Q8. What is the significance of the rescaling of the plane?
In particular it happens (see [5]) that all double bubbles (i.e. theminimal clusterswith two regions, see [2])can be obtained, up to rescaling, by projecting the geodesic network on the sphere given by three meridiansjoining in two antipodal points with equal angles of 120 degrees.
Q9. What is the vertices of the Releaux triangle?
Notice that the Releaux triangle with vertices z 0 , z 1 , z 2 is increasing (in the sense of set inclusion) with respect to the parameter ρ.
Q10. What is the way to determine the existence of a minimizer in R2?
Existence and regularity of minimizers inR2 was proved byMorgan [6] (see also [4]): the regions of a minimizer inR2 are delimited by a nite number of circular arcs which meet in triples at their end-points.
Q11. what is the area of the upper region of the cluster?
The area of the upper region E 1 is given by|E 1 | = 2A(w 0 , w 2 , β 0 − 2π/3)A(w 2 , w 5 , β 1 ) − A(w 0 , w 3 , β 0 )and the area of the lower region E 2 is given by|E 2 | = −2A(w 0 , w 1 , β 0 + 2π/3) − A(w 1 , w 4 , β 1 ) + A(w 0 , w 3 , β 0 ).
Q12. What is the symmetry of the vertices of the Releaux triangle?
This means that the minimal cluster E is itself symmetric with respect to the axis x = 1/2 and the left hand side triangular region E 4 can be obtained by symmetry once E 3 has been determined.
Q13. What is the formula for calculating the area of the cluster E?
This allows us to nd ρ = ρ(θ) such that |E 3 | = |E 2 | using a simple bisection method since |E 3 | − |E 2 | is decreasing in ρ.
Q14. What is the simplest way to determine the area of the cluster?
Theorem 3.1. Let E = (E 1 , E 2 , E 3 , E 4 ) be a cluster with N = 4 regions in the plane which is minimal with prescribed equal areas a = (a, a, a, a).