GEORGIAN MATHEMATICAL JOURNAL: Vol. 3, No. 3, 1996, 233-238

SOME PARTITIONS CONSISTING OF JORDAN CURVES

A. KHARAZISHVILI

Abstract. Some decompositions of the three-dimensional sphere and three-dimensional ball into Jordan curves are considered. In particular, it is proved that for every strictly positive real number p ≤ 1 there exists a partition of the unit three-dimensional Euclidean sphere into circles whose radii are equal to p.

Let S^{2} be the unit two-dimensional sphere in the Euclidean space ℝ^{3} and let S^{3} be the unit three-dimensional sphere in the Euclidean space ℝ^{4}. The well-known Hopf fibration φ : S^{3} → S^{2} can be expressed analytically as

φ(x_{1}, x_{2}, x_{3}, x_{4}) = (z_{1}, z_{2}, z_{3}),

where we have

z_{1} = 2(x_{1}x_{3} + x_{2}x_{4}),
z_{2} = 2(x_{2}x_{3} − x_{1}x_{4}),
z_{3} = x_{1}^{2} + x_{2}^{2} − x_{3}^{2} − x_{4}^{2}.

Moreover, if (z_{1}, z_{2}, z_{3}) is any point of the sphere S^{2} and a point (x_{1}, x_{2}, x_{3}, x_{4}) belongs to the set φ^{−1}((z_{1}, z_{2}, z_{3})), then φ^{−1}((z_{1}, z_{2}, z_{3})) can be expressed as the set of all points (y_{1}, y_{2}, y_{3}, y_{4}) ∈ S^{3} which satisfy the equalities

\begin{matrix}y_{1} & = x_{1} cos θ − x_{2} sin θ, \\ y_{2} & = x_{1} sin θ + x_{2} cos θ, \\ y_{3} & = x_{3} cos θ − x_{4} sin θ, \\ y_{4} & = x_{3} sin θ + x_{4} cos θ. \end{matrix} (0 ≤ θ ≤ 2π)

From these equalities one can see that for any point (z_{1}, z_{2}, z_{3}) ∈ S^{2} the preimage φ^{−1}((z_{1}, z_{2}, z_{3})) is a circle of radius 1, lying on the plane of ℝ^{4}

233 generated by the following two vectors:

1991 Mathematics Subject Classification. 54F50.

Key words and phrases. Partitions, Jordan curve, Euclidean sphere, Euclidean ball, Hopf fibration.

1072-947X/96/0500-0233$09.50/0 \begin{matrix}c \\ ◯ \end{matrix} 1996 Plenum Publishing Corporation

(x_{1}, x_{2}, x_{3}, x_{4}), (−x_{2}, x_{1}, −x_{4}, x_{3}).

In other words, the Hopf fibration gives us a set-theoretic partition of the unit sphere S^{3} into circles of radii 1 and it is well known that this partition has a number of interesting geometrical and topological properties.

Now, let p be a real number such that 0 < p < 1. A natural question arises: does there exist a partition (in the set-theoretic sense) of the sphere S^{3} into circles of radii p? It turns out that such a partition always exists. In order to prove this fact (and a more general result) we need the following auxiliary assertion.

Lemma 1. Let c be the cardinality of the continuum and let (L_{j})_{j∈J} be a family of circles in the space ℝ^{4} such that card(J) < c. Let x be an arbitrary point of the sphere S^{3} not belonging to the set ∪_{j∈J} L_{j}. Then for

every strictly positive real number p < 1 there exists a circle L satisfying the conditions:

(1) L ⊂ S^{3},

(2) x ∈ L,

(3) the radius of L is equal to p,

(4) L does not intersect the set ∪_{j∈J} L_{j}.

Proof. For any index j ∈ J let P_{j} be the affine plane in the space ℝ^{4} containing the circle L_{j}. Since we have the inequality card(J) < c, there exists a hyperplane Γ in the space ℝ^{4} passing through the origin and such that

x ∈ Γ, (∀j ∈ J)(dim(P_{j} ∩ Γ) ≤ 1).

Let us consider the set Γ∩S^{3}. Obviously, this set is the unit two-dimensional sphere in the three-dimensional space Γ. Moreover, we have

(∀j ∈ J)(card(Γ ∩ S^{3} ∩ L_{j}) ≤ 2).

Let us put

Z = Γ ∩ S^{3} ∩ ( ⋃_{j∈J} L_{j}).

Then we have

card(Z) < c, x ∉ Z.

Now, it is clear that there exists a circle L ⊂ Γ ∩ S^{3} such that its radius is equal to p and

x ∈ L, L ∩ Z = ∅

Obviously, the circle L is a required one.

With the help of Lemma 1 we can obtain the following theorem.

Theorem 1. Let I be a set of cardinality c and let (p_{i})_{i∈I} be a family of real numbers such that

(∀i ∈ I)(0 < p_{i} < 1).

Then there exists a partition (L_{i})_{i∈I} of the unit sphere S^{3} satisfying the conditions:

(1) (∀i ∈ I) (L_{i} is a circle),

(2) (∀i ∈ I) (the radius of L_{i} is equal to p_{i}).

Proof. Let α be the least ordinal number of cardinality c. Without loss of generality we may assume that

I = {ξ : ξ < α}.

Let (x_{ξ})_{ξ<α} be an injective family of all points of the unit sphere S^{3}. Let us construct by the transfinite recursion a family (L_{ξ})_{ξ<α} of circles satisfying the following relations:

(a) for any ζ < ξ < α we have L_{ζ} ∩ L_{ξ} = ∅,

(b) for each ξ < α the radius of the circle L_{ξ} is equal to p_{ξ},

(c) for each ξ < α the point x_{ξ} belongs to the set ∪_{ζ≤ξ}L_{ζ}.

Suppose that for an ordinal number ξ < α a partial family of circles (L_{ζ})_{ζ<ξ} has already been defined. Of course, we have

S^{3}\ ⋃_{ζ<ξ} L_{ζ} ≠ ∅.

Let y be an arbitrary point from the set S^{3}\ ∪_{ζ<ξ} L_{ζ}. Let us put

x = \begin{cases}x_{ξ} & if x_{ξ} ∉ ⋃_{ζ<ξ} L_{ζ}, \\ y & if x_{ξ} ∈ ⋃_{ζ<ξ} L_{ζ}. \end{cases}

Then we can apply Lemma 1 to the point x and to the family of circles (L_{ζ})_{ζ<ξ}. By this lemma there exists a circle L ⊂ S^{3} such that its radius is equal to p_{ξ} and

x ∈ L, L ∩ ( ⋃_{ζ<ξ} L_{ζ}) = ∅.

Let us put L_{ξ} = L. It is clear that in this way we shall be able to construct the required partition (L_{ξ})_{ξ<α} of the unit sphere S^{3}.

Remark 1. If for every i ∈ I we have p_{i} = p, then we obtain a partition of the sphere S^{3} into pairwise congruent circles (all of radius p). It is interesting to investigate for which real numbers p there exists an effective partition of the sphere S^{3} into circles having radii p (“effective” means here that the axiom of choice will not be used). The example of the Hopf fibration φ shows us that for p = 1 an effective partition of S^{3} into circles with radii p does exist. It is worth observing here that by the method described above we can create many (non-effective) partitions of the sphere S^{3} into circles with radii 1. Thus we see that there exists a partition of S^{3} into circles with radii 1, essentially different from the Hopf partition. Of course, these partitions do not have some good geometrical or topological properties as the Hopf partition. Notice also that there exists a curve L satisfying the following three relations:

(1) L is homeomorphic to S^{1};

(2) L is contained in S^{3};

(3) there does not exist a partition of S^{3} into curves congruent to L.

In connection with the mentioned fact see [1], where an analogous proposition is established for the three-dimensional Euclidean space ℝ^{3}.

Moreover, in [1] the following problem is posed: give a characterization of all Jordan curves L lying in ℝ^{3} such that there exists a partition of ℝ^{3} into curves congruent to L. Clearly, a similar problem can be formulated for the three-dimensional sphere S^{3}.

These two problems remain open.

Remark 2. Let I be an arbitrary set of cardinality c and let (p_{i})_{i∈I} be an arbitrary family of strictly positive real numbers. By the method described above one can prove the existence of a partition (L_{i})_{i∈I} of the space ℝ^{3} into circles such that

(∀i ∈ I)(L_{i} has the radius p_{i}).

In particular, we see that there exists a partition of the space ℝ^{3} into pairwise congruent circles. A similar result is true for the unit closed threedimensional ball B_{3} ⊂ ℝ^{3} (see [1]).

For our further consideration we need the following

Lemma 2. Let S^{2} be the unit sphere in the Euclidean space ℝ^{3} and let B_{3} be the closed unit ball in the same space. Let (L_{j})_{j∈J} be a family of curves in ℝ^{3} satisfying the conditions:

(1) card(J) < c,

(2) (∀j ∈ J) (L_{j} is an arc of a circle). Then for any two distinct points x ∈ S^{2} and y ∈ S^{2} there exists a curve L such that

(a) L is an arc of a circle,

(b) the set of end-points of L is {x, y},

(c) L ⊂ B_{3},

(d) L ∩ S^{2} = {x, y},

(e) (∀j ∈ J) (L_{j} ∩ L ⊆ {x, y}).

Proof. The reasoning is quite similar to the arguments used in the proof of Lemma 1. For any index j ∈ J let P_{j} be the plane in the space ℝ^{3} containing the arc L_{j}. Since we have the inequality

card(J) < c,

there exists a plane Γ in the space ℝ^{3} such that

x ∈ Γ, y ∈ Γ, (∀j ∈ J)(dim(P_{j} ∩ Γ) ≤ 1).

Hence we obtain

(∀j ∈ J)(card(L_{j} ∩ Γ) ≤ 2).

Let us put

Z = Γ ∩ ( ⋃_{j∈J} L_{j}).

Obviously, we have

card(Z) < c.

Now, it is clear that in Γ ∩ B_{3} there exists an arc of the circle L such that x and y are end-points of L and

L ∩ S^{2} = {x, y},
L ∩ Z ⊆ {x, y}.

This arc L is a required one.

Theorem 2. Let S^{2} be the unit two-dimensional sphere in the space ℝ^{3} and let B_{3} be the closed unit ball in the same space. Then there exists a family (L_{i})_{i∈I} of curves satisfying the following conditions:

(1) (∀i ∈ I)(L_{i} is an arc of a circle),

(2) (∀i ∈ I) (L_{i} ⊂ B_{3}),

(3) (∀i ∈ I) (L_{i} ∩ S^{2} is the set of end-points of L_{i}),

(4) (∀i ∈ I)(∀i′ ∈ I)(i ≠ i′ ⇒ L_{i} ∩ L_{i′} ⊂ S^{2}),

(5) for any two distinct points x ∈ S^{2} and y ∈ S^{2} there exists an arc L_{i} such that {x, y} is the set of end-points of L_{i}.

Proof. Let α be the least ordinal number of cardinality c. Let ({x_{ξ}, y_{ξ}})_{ξ<α} be an injective family of all two-point subsets of the sphere S^{2}. Let us construct the required family of arcs (L_{ξ})_{ξ<α} by the method of transfinite recursion. Suppose that for an ordinal number ξ < α a partial family (L_{ζ})_{ζ<ξ} has already been defined. For any ζ < ξ let X_{ζ} be the set of end-points of the arc L_{ζ}. Consider the family (X_{ζ})_{ζ<ξ}. Let η be the least ordinal number such that

{x_{η}, y_{η}} ∉ (X_{ζ})_{ζ<ξ}.

We can apply Lemma 2 to the points x_{η}, y_{η} and to the family of arcs (L_{ζ})_{ζ<ξ}. By this lemma there exists an arc of the circle L_{ξ} satisfying relations analogous to relations (a)–(e) of the lemma. Hence, in this way we shall be able to construct the family of arcs (L_{ξ})_{ξ<α}. Now, if we put

I = {ξ : ξ < α},

then it is easy to check that the family (L_{i})_{i∈I} satisfies conditions (1)–(5) of Theorem 2.

Remark 3. Slightly changing the proof of Theorem 2 presented above, we can find a family of arcs (L_{i})_{i∈I} satisfying conditions (1)–(5) of this theorem and also satisfying the following condition:

(6) for each point x ∈ B_{3} there exists an arc L_{i} such that x ∈ L_{i}. It is reasonable to notice here that we can also prove a result analogous to Theorem 2 for a family (L_{i})_{i∈I} consisting of polygonal lines. Moreover, we may even assume that any polygonal line L_{i} (i ∈ I) consists at most of two segments.

A natural question arises: is it possible to prove Theorem 2 effectively (i.e., without using the axiom of choice)?

Finally, notice that the formulations of both Theorems 1 and 2 are, in fact, due to elementary geometry of the Euclidean space, but the proof of these theorems requires essentially non-elementary methods.

References

1. A. B. Kharazishvili, Applications of set theory. (Russian) Tbilisi University Press, Tbilisi, 1989.

(Received 07.09.1994)

Author’s address:

I. Vekua Institute of Applied Mathematics

Tbilisi State University

2, University St., Tbilisi 380043

Republic of Georgia