First of all, we shall define a space of n-simplices, upon which flows of n-simplices are defined. The topology of the space is defined using “adjacent” relationship between n-simplices.

Definition 1 (n-simplex). Let n ∈ ℕ . An n-simplex is the convex hull of ( n + 1 ) affinely independent points in E n (i.e., points not lying in a ( n − 1 ) -dimensional subspace). The convex hull of n + 1 points v 0 , v 1 , ⋯ , v n ∈ E n is denoted by [ v 0 , v 1 , ⋯ , v n ] , i.e.,

[ v 0 , v 1 , ⋯ , v n ] : = { ∏ i = 0 , 1 , ⋯ , n v i λ i ∈ E n | ∑ i = 0 , 1 , ⋯ , n λ i = 1     and     ∀ i , λ i ≥ 0 } .

Then, v i ( 0 ≤ i ≤ n ) are called the vertices of [ v 0 , v 1 , ⋯ , v n ] . Let s be an n-simplex. The set of all the vertices of s is denoted by v ( s ) .

For example, a 0-simplex is a point, a 1-simplex is a line segment, a 2-simplex is a triangle, a 3-simplex is a tetrahedron.

Definition 2 (k-face). Let k , n ∈ ℕ and k ≤ n . Let s be an n-simplex. A k-face of s is the convex hull of any k + 1 vertices of s. A 0-face is a vertex of s. A 1-face is called an edge of s. An ( n − 1 ) -face is called a facet of s. Note that the n-face is s itself.

For example, let s = [ v 0 , v 1 , v 2 , v 3 ] be a tetrahedron. Then, s has 6 edges [ v i , v j ] ( 0 ≤ i < j ≤ 3 ) and 4 facets [ v i , v j , v k ] ( 0 ≤ i < j < k ≤ 3 ). Moreover, v ( s ) = { v 0 , v 1 , v 2 , v 3 } .

Definition 3 (n-simplex space). Let M be a set of n-simplices. M is called an n-simplex space if each n-simplex is connected to other n-simplices in such a way that,

for ∀ facet u of s ∈ M , uniquely s ′ ∈ M such that s ∩ s ′ = u .

In particular, each n-simplex is connected to n + 1 “adjacent” n-simplices through its n + 1 facets.

Example 1. We would obtain an n-simplex space by partitioning E n into pieces of n-simplices. Shown in Figure 1(a) is a triangle space M 0 obtained by partitioning E 2 into pieces of triangles.

Definition 4 (k-face neighborhood N(u)). Let M be an n-simplex space and s ∈ M . Let u be a k-face of s. The k-face neighborhood N ( u ) of u is a set of n-simplices of M which contain u:

N ( u ) : = { s ′ ∈ M | u ⊂ s ′ } .

For s = [ v 0 , v 1 , ⋯ , v n ] ∈ M , we obtain s = ∩ i = 0 , 1 , ⋯ , n   N ( v i ) . Note that every facet neighborhood consists of two n-simplices. We shall use the fact when defining local trajectories of n-simplices (See just above Definition 6).

Now let us define flows of n-simpleces on an n-simplex space.

Definition 5 (Tangent space T(s)). Let s be an n-simplex. The tangent space T ( s ) at s is the set of all the edges of s, i.e.,

T ( s ) : = { [ v j , v j ] | 0 ≤ i < j ≤ n } ,

where v ( s ) = { v 0 , v 1 , ⋯ , v n } . A subset of T ( s ) is called a gradient of s.

Example 2. In Figure 1(b), a gradient (i.e., a set of edges) is assigned to each triangle of M 0 . Most of the triangles are assigned one edge, some are assigned multiple edges, and others are assigned no edge. Shown in Figure 1(c) are all the possible values of the gradient of a triangle s 0 of M 0 , from which the encircled value is assigned to s 0 in (b).

Let M be an n-simplex space. Let s = [ v 0 , v 1 , ⋯ , v n ] ∈ M and e = [ v a , v b ] ∈ T ( s ) . Two facets u a ( s , e ) and u b ( s , e ) of s which do not contain the edge e are defined by

{ u a ( s , e ) = [ v 0 , ⋯ , v a ^ , ⋯ , v n ] , u b ( s , e ) = [ v 0 , ⋯ , v b ^ , ⋯ , v n ] ,

where ^ means that the corresponding term is omitted.

Then, by definition, there are two n-simplices s a ( s , e ) and s b ( s , e ) ∈ M such that

{ N ( u a ( s , e ) ) = { s , s a ( s , e ) } , N ( u b ( s , e ) ) = { s , s b ( s , e ) } .

Definition 6 (Adjacent n-simplices A(s,G)). Let M be an n-simplex space. Let s ∈ M and e ∈ T ( s ) . The adjacent n-simplices A ( s , e ) associated with the edge e of s is defined by

A ( s , e ) : = { s a ( s , e ) , s b ( s , e ) } ,

where s a ( s , e ) and s b ( s , e ) are defined above. That is, A ( s , e ) is the set of all the adjacent triangles of s which do not contain the edge e.

Let G ⊂ T ( s ) be a gradient of s. The adjacent n-simplices A ( s , G ) associated with G is defined by

A ( s , G ) : = ( ∩ e ∈ G A ( s , e ) , if   G ≠ ∅ all the adjacent n -simplices of s , if   G = ∅

In particular A ( s , e ) = { s ′ ∈ A ( e , ∅ ) | e ⊂ s ′ } .

Definition 7 (Local trajectory at an n-simplex). Let M be an n-simplex space. Let s ∈ M and e ∈ T ( s ) . Let A ( s , e ) = { s a , s b } . The local trajectory at s associated with the edge e is the sequence

s a − s − s b   ( or   s b − s − s a )

of three consecutive n-simplices. Connecting these sequences together, we shall obtain a flow on M in Definition 12 and 13.

Example 3. Grey triangles in Figure 1(c) are the adjacent triangles A ( s 0 , G ) associated with the gradient G (thick lines) of s 0 .

Conversely, a sequence s 0 − s − s 2 of three consecutive n-simplices determines uniquely an edge of the middle n-simplex s as follows.

Definition 8 (Tangent D t ( s 0 − s − s 2 ) ). Let M be an n-simplex space. Let s 0 − s − s 2 be a sequence of three consecutive n-simplices of M, i.e., s 0 , s 2 ∈ A ( s , ∅ ) such that s 0 ≠ s 2 . Let

{ u 0 : = s ∩ s 0   ( the facet shared by s and s 0 ) , u 2 : = s ∩ s 2   ( the facet shared by s and s 2 ) .

The tangent D t ( s 0 − s − s 2 ) to s 0 − s − s 2 at s is an edge [ v 0 , v 2 ] of s, where

{ v 0 : = v ( s ) \ v ( u 0 )   ( the vertex not included in u 0 ) , v 2 : = v ( s ) \ v ( u 2 )   ( the vertex not included in u 2 ) .

Note that D t is not defined at singular simplices because singular simplices never occupy the middle position of a sequence of three consecutive n-simplices. (See Figure 1(c).)

Lemma 1. Let M be an n-simplex space. Let s ∈ M and e ∈ T ( s ) . Let s 0 − s − s 2 be a sequence of three consecutive n-simplices of M. Then,

{ A ( s , D t ( s 0 − s − s 2 ) ) = { s 0 , s 2 } , D t ( s a ( s , e ) − s − s b ( s , e ) ) = D t ( s b ( s , e ) − s − s a ( s , e ) ) = e ,

where s a ( s , e ) and s b ( s , e ) are the two n-simplices of A ( s , e ) .

Proof. It follows immediately from the definition.

A differential structure is defined on an n-simplex space as follow.

Definition 9 (Tangent bundle ( T M , M , π M ) ). Let M be an n-simplex space. The tangent bundle ( T M , M , π M ) of M is defined by

{ T M : = { ( s , u ) | s ∈ M , u ∈ T ( s ) } , π M : T M → M ,   π ( s , u ) : = s .

Definition 10 (Vector field V on M). Let M be an n-simplex space. A vector field V on M is a mapping which assigns to each n-simplex s of M, a gradient of s, i.e.,

V : M → 2 T ( s ) , V ( [ v 0 , v 1 , ⋯ , v n ] ) = { [ v i , v j ] , ⋯ , [ v k , v l ] } ,

where 2 T ( s ) denotes the power set of T ( s ) . If V ( s ) contains only one edge, s is called a regular n-simplex of V. Otherwise, s is called a singular n-simplex of V. If V ( s ) = ∅ , s is called a branch n-simplex of V. If V ( s ) consists of m edges, s is called an m-fold singular n-simplex of V. If A ( s , V ( s ) ) has only one n-simplex, s is called a terminal n-simplex of V. If A ( s , V ( s ) ) = ∅ , s is called an isolated n-simplex.

Example 4. Shown in Figure 1(b) is a vector field of the triangle space M 0 of (a).

Definition 11 (Local trajectory of V on M). Let M be an n-simplex space and s ∈ M . Let V be a vector field of M. Let s 0 − s − s 2 be a sequence of three consecutive n-simplices. Then, s 0 − s − s 2 is called a local trajectory of V at s if

D t ( s 0 − s − s 2 ) ⊃ V ( s ) .

Note that local trajectories may contain branch n-simplices.

Definition 12 (Trajectory of V on M). Let M be an n-simplex space. Let V be a vector field of M. Let L = { s [ i ] | i ∈ I ⊂ ℤ } be a sequence of n-simplices, where I is either [ k , m ] , [ k , + ∞ ) , ( − ∞ , m ] , or ( − ∞ , + ∞ ) ( k , m ∈ ℤ such that k < m ). Then, L is called a trajectory of V if every consecutive three n-simplices of L is a local trajectory of V. i.e.,

s [ i ] − s [ i + 1 ] − s [ i + 2 ] isalocaltrajectoryof V   for ∀ [ i , i + 2 ] ⊂ I .

A trajectory L = { s [ i ] | i ∈ [ k , m ] ⊂ ℤ } of V is called closed if

s [ m − 1 ] − s [ m ] − s [ k ]     and     s [ m ] − s [ k ] − s [ k + 1 ]

are also local trajectories of V.

A trajectory L of V is called maximal if either L is closed, or L ′ ⊃ L implies L ′ = L for any trajectory L ′ of V on M.

Definition 13 (Flow of V on M). Let M be an n-simplex space. Let V be a vector field of M. Let F = { L i | i ∈ I ⊂ ℤ } be a set of maximal trajectories of V on M, where L i ≠ L j if i ≠ j . Then, L is called a flow of V on M if

M = ∪ i ∈ I L i   .

Note that M is decomposed into a disjoint union of L i if V has no branch triangle.

Here we define two functions on trajectories of vector fields on an n-simplex space.

Definition 14 (U/D function g along a trajectory). Let M be an n-simplex space. Let V be a vector field of M. Let L = { s [ 0 ] , s [ 1 ] , ⋯ , s [ k ] } ( k ∈ ℕ ) be a trajectory of V on M. An U/D function g along L is a { + 1, − 1 } -valued function on L defined by

{ g ( s [ 0 ] ) ∈ { + 1, − 1 } ⊂ ℤ , g ( s [ i + 1 ] ) : = ( g ( s [ i ] ) , if   V ( s [ i + 1 ] ) ∩ V ( s [ i ] ) = ∅ − g ( s [ i ] ) ,                         otherwise

Example 5. Shown in Figure 2 is a trajectory L a = { s [ 0 ] , s [ 1 ] , s [ 2 ] , s [ 3 ] , s [ 4 ] , ⋯ } of the vector field of M 0 given in Figure 1(b), where s [ 0 ] = s 0 . Then, we obtain an U/D function g a along L a as follows: Firstly, set g a ( s [ 0 ] ) = + 1 and move to the adjacent triangle s [ 1 ] on the right. Then, g a ( s [ 1 ] ) = − g a ( s [ 0 ] ) = − 1 since V ( s [ 1 ] ) ∩ V ( s [ 0 ] ) ≠ ∅ . In the same way, we obtain g a ( s [ 2 ] ) = − g a ( s [ 1 ] ) = + 1 . Now, let us move to s [ 3 ] . Then, g a ( s [ 3 ] ) = g a ( s [ 2 ] ) = + 1 since V ( s [ 3 ] ) ∩ V ( s [ 2 ] ) = ∅ . In the same way, we obtain g a ( s [ 4 ] ) = g a ( s [ 3 ] ) = + 1 .

By considering the “integral along the trajectory” of a given U/D function, we shall obtain another function on the trajectory.

Definition 15 (Height function h g on a trajectory). Let M be an n-simplex space. Let V be a vector field of M. Let L = { s [ 0 ] , s [ 1 ] , ⋯ , s [ k ] } ( k ∈ ℕ ) be a trajectory of V on M. Let g be a U/D function along L. The height function h g with respect to g is a ℤ -valued function on L defined by

{ h g ( s [ 0 ] ) ∈ ℤ , h g ( s [ i + 1 ] ) : = ( h g ( s [ i ] ) + g ( s [ i ] ) , if   V ( s [ i + 1 ] ) ∩ V ( s [ i ] ) = ∅ h g ( s [ i ] ) ,                     otherwise

Example 6. Shown in Figure 2 is a trajectory of the vector field of given in Figure 1(b), where. The table on the right shows the values of an U/D function along (See Example 5) and the height function with respect to. is obtained as follows: Firstly, set. Since , we obtain. In the same way, we obtain. Now, let us move to. Then, and

In the same way, we obtain.

Shown in Figure 3(a) is a triangle space obtained by partitioning into triangles of the same shape. A vector field on is shown in Figure 3(b). In this case, the “two-dimensional” vector field of corresponds to a “three-dimensional” drawing on the surface of “mountains” of unit cubes of as shown in Figure 3(c) and Figure 3(f). It is this type of vector fields of triangles that is considered in this section.

Definition 16 (The three-dimensional lattice). Let be the three-dimensional lattice generated by three vectors, , and, i.e.,

Shown in Figure 3(d) is a unit cube of and its top view.

Definition 17 (The symmetric group on three letters). Let be the group of all the permutations of the set. Elements of are written in cyclic notation. For example, let. Then, , , and.

Definition 18 (The set B₂ of all flat triangles). Let and. The slant triangle is defined by

where denotes the convex hull of three points (Definition 1). For example, the four slant triangles shown in Figure 3(e) are, , , and (from top to bottom).

Let be the set of all slant triangles, i.e.,

The shift operator on is defined by

Then, an equivalence relation is defined on by

if and only if

The -equivalence class of is called a flat triangle and denoted by. For example, shown in Figure 3(e) is the -equivalence class of ().

The set of all flat triangles is denoted by, i.e.,

Lemma 2. is a triangle space (Definition 3).

Proof. It follows immediately from the definition.

By an abuse of notation, the “image on “ of an edge e of is also denoted by. Note that

where and (See Figure 3(e)). The tangent space at (Definition 5) is given by

where and.

For simplicity, we often identify the edge with the monomial and we shall obtain a one-to-one correspondence

Definition 19 (Tangent bundle). The tangent bundle of (Definition 9) is given by

Definition 20 (Gradient). Let. The gradient of t is defined by

That is, is a -valued function on. The “edge” is also called the boundary edge of. (Strictly speaking, is a set of one element. Here, we identify the set with its only element.)

Example 7. In Figure 3, the boundary edges are drawn with a thick line. For example, the boundary edge of is (Figure 3(d)).

Lemma 3. Let. Then,

That is, induces a -valued function on. By an abuse of notation, the induced function is also denoted by, i.e.,

Proof. By definition,

where.

Lemma 4. Let. Then, the local trajectory at associated with (Definition 7) is either

or

where

(See Figure 4(a)). The local trajectory is called the local trajectory associated with.

Proof. Note that the two facets which do not contain the boundary edge are

(See above Definition 6). The result follows immediately.

Now, let us give the definition of “mountains of unit cubes” shown in Figure 3(c) and Figure 3(f).

Definition 21 (A tangent cone Cone A). Let A be a finite subset of. A three-dimensional tangent cone is defined by

The set of all the slant triangles on the surface of is denoted by , i.e.,

Example 8. The tangent cone corresponding to the “mountains of unit cubes” of Figure 3(c) and Figure 3(f) is given by

where, , , , , , , , , , , , and.

Lemma 5. Let be a tangent cone. Then,

where

Proof. For, s.t.. Then, is the coordinate of p with respect to “origin” a. In particular,

The result follows immediately.

The surface of a tangent cone induces a vector field of.

Definition 22 (Vector field on). Let be a tangent cone. The vector field on induced by c is defined by

is called a -embeddable vector field of triangles. Note that has no singular triangle.

Remark 1., and the value of is determined uniquely on.

Example 9. Shown in Figure 3(c) and Figure 3(f) is the vector field on induced by of Example 8.

Local trajectories of on (Definition 11) is computed as follows.

Lemma 6. Let. Let be a tangent cone. Suppose that . Then, the local trajectory of at s is either

or

where

Proof. See Lemma 4.

Lemma 7. Let. Let be a tangent cone. Let be the local trajectory of at s. Then,

Proof. Note that has no singular triangle on. The result follows immediately.

Proposition 1. Let V be a vector field on without singular triangles. Then,

Proof. See [4].

Vector fields induced by a tangent cone are inherently associated with an U/D function and a height function.

Let. Let be a tangent cone. Suppose that . Then, the local trajectory at s is either or, where

(See Figure 4(a)).

Definition 23 (U/D function). Let. Let be a tangent cone. Let be the local trajectory of at s (i.e.). The U/D function at s along the trajectory associated with is defined by

where and are given above. That is, −1 and +1 indicate “downhill” and “uphill” on the “mountain road”, respectively.

Remark 2. In Definition 14, U/D functions are not uniquely specified on an n-simplex space because the uphill and downhill along a trajectory are not given explicitly. On the other hand, the U/D function is uniquely specified on using the uphill and downhill along a trajectory of slant n-simplices.

Lemma 8. Let be a tangent cone. Then, is an U/D function defined in Definition 14.

Proof. Let be a local trajectory of. Let and. Suppose that. Then, either

or

Suppose that. Then, , where either

or

respectively. Since, we obtain

That is,

Continuing in the same way for the other case, we obtain

The result follows immediately.

Proposition 2. Let be a tangent cone. Let be a maximal trajectory of on. Let. Then,

Remark 3. The edge neighborhood consists of two triangles which share the boundary edge (Definition 4).

Proof. Let (). Suppose that . Then, either or is enclosed by the trajectory of finite length, and the trajectory starting from the enclosed triangle (either or ) has an “end point”. However, has no singular triangle, which is a contradiction.

Corollary 1. Suppose that L is closed. Then, the sum of over the “boundary” of L is equal to zero, i.e.,

where.

Proof. Because of Proposition 2, the sum of over L is equal to the sum of over the “boundary” of L, i.e.,

where. Since the sum of over L is zero when L is closed, the result follows.

Example 10. Shown in Figure 4(b) above is the value of the U/D function along a trajectory shown in Figure 3(b). The grey triangles belong to, and the white triangles belong to. The value of on the first three triangles are

Since

we obtain. Since

we obtain.

Note that two grey triangles sharing a thick edge have opposite values. The sum of the U/D function over the set of all the white triangles is equal to zero.

Definition 24 (Height function). The height function on is a -valued function defined by

The height function on is a -valued function defined by

Let be a tangent cone. Then, the height function on associated with is a -valued function defined by

where.

By an abuse of notation, we use the same name for three functions with different domains.

Lemma 9. Let be a tangent cone. Then, is a height function with respect to defined in Definition 15.

Proof. Let be a local trajectory of. Then, on is given by

The result follows immediately.

Proposition 3. Let. Let be a tangent cone. Then, is constant on, i.e.,

Proof. Since, the result follows immediately.

Example 11. Shown in Figure 4(b) below is the value of the height function along a trajectory shown in Figure 3(b). The value of on the first three triangles are

where, ,. Then,

In the same way, we obtain.

Note that two grey triangles sharing a thick edge have the same value.

This paper proposes a novel mathematical approach for the design of self-assembling molecules, where self-assembling molecules are represented as a union of trajectories of tetrahedrons. Here we shall consider vector fields on a tetrahedron space which are induced by a four-dimensional tangent cone.

In the same way as for the space of flat triangles, we shall define a “tetrahedron space” by partitioning into tetrahedrons of the same shape. “Three-dimensional” vector fields of tetrahedrons then correspond to a “four-dimensional” drawing on the surface of “mountains” of unit cubes of.

Definition 25 (The four-dimensional lattice). Let be the four-dimensional lattice generated by four vectors, , , and, i.e.,

Shown in Figure 5(a) is a unit cube of and its “top view”.

Definition 26 (The symmetric group on four letters). Let be the group of all the permutations of the set. Elements of are written in cyclic notation. For example, let. Then, , , , and.

Definition 27 (The set B₃ of all flat tetrahedrons). Let and. The slant tetrahedron is defined by

where denotes the convex hull of four points (Definition 1).

Let be the set of all slant tetrahedrons, i.e.,

The shift operator on is defined by

Then, an equivalence relation is defined on by

The -equivalence class of is called a flat tetrahedron and denoted by.

The set of all flat tetrahedrons is denoted by, i.e.,

Example 12. The facet of a unit cube shown in Figure 5(b) bottom consists of six slant tetrahedrons

For example, is the tetrahedron. Then, the “projection image” of the facet is divided into six flat tetrahedrons (Figure 5(b) top)

Note that all the tetrahedrons share the “diagonal” edge.

Lemma 10. is a tetrahedron space (Definition 3).

Proof. It follows immediately from the definition.

By an abuse of notation, the “image on” of a k-face u of is also denoted by. The tangent space at (Definition 5) is then given by

where, and.

For simplicity, we often identify the edge and with the monomial and, respectively. Then, we shall obtain a one-to-one correspondence

Definition 28 (Tangent bundle). The tangent bundle of (Definition 9) is given by

Definition 29 (Gradient D_St). Let. The gradient of t is defined by

That is, is a -valued function on. In particular,

The “edge” is called the boundary edge of. (Strictly speaking, is a set of one element. Here, we identify the set with its only element.)

Example 13. Shown in Figure 5(c) is the flat tetrahedron (top) and its six edges (thick lines, bottom). All the six edges are shown with the adjacent tetrahedrons associated with them (Definition 6). Only four of them are included in the image of (left and center). Roughly speaking, U-turns are prohibited on (right).

Lemma 11. Let. Then,

That is, induces a -valued function on. By an abuse of notation, the induced function is also denoted by, i.e.,

Proof. It can be proved in the same way as the proof of Lemma 3.

Lemma 12. Let. Then, the local trajectory at associated with (Definition 7) is

or

where

The local trajectory is called the local trajectory associated with.

Proof. Note that the two facets which do not contain the edge are

(See above Definition 6). The result follows immediately.

Example 14. Shown in Figure 5(d) (right) is a trajectory on obtained by patching overlapping local trajectories together. By connecting tetrahedrons via edges and, we shall obtain a chain of isosceles tetrahedrons as shown on the left. In the case of, trajectories are obtained by folding the chain of tetrahedrons.

Four-dimensional “mountains of unit cubes” is defined as follows.

Definition 30 (A tangent cone Cone A). Let A be a finite subset of. A four-dimensional tangent cone is defined by

The set of all the slant tetrahedrons on the surface of is denoted by, i.e.,

Lemma 13. Let be a tangent cone. Then,

where

Proof. It follows immediately from the definition.

The surface of a tangent cone induces a vector field of.

Definition 31 (Vector field on). Let be a tangent cone. The vector field on induced by c is defined by

is called a -embeddable vector field of tetrahedrons. Note that has no singular tetrahedron.

Local trajectories of on (Definition 11) is computed as follows.

Lemma 14. Let. Let be a tangent cone. Suppose that . Then, the local trajectory of at s is either

or

where

Proof. See Lemma 12.

Example 15. Shown in Figure 5(e) are four closed trajectories of the vector field on induced by

(The fourth trajectory is hidden behind others). A rhombic dodecahedron is divided into the set of four closed trajectories of length six.

There is no proof of the following claim.

Problem 1. Let V be a vector field on without singular tetrahedrons. Then, show that

Vector fields induced by a tangent cone are inherently associated with an U/D function and a height function.

Definition 32 (U/D function). Let. Let be a tangent cone. Let be the local trajectory of at s (i.e.). The U/D function at s along the trajectory associated with is defined by

where and are given in Lemma 14.

Lemma 15. Let be a tangent cone. Then, is an U/D function defined in Definition 14.

Proof. It can be proved in the same way as the proof of Lemma 8.

There is no proof of the following two claims (See Proposition 2).

Problem 2. Let be a tangent cone. Let L be a maximal trajectory of on. Let. Then, show that

Problem 3. Let be a tangent cone. Let L be a maximal trajectory of on. Suppose that L is closed. Let. Then, show that the sum of over the “boundary” of L is equal to zero, i.e.,

where.

Definition 33 (Height function). The height function on is a -valued function defined by

The height function on is a -valued function defined by

Let be a tangent cone. Then, the height function on associated with is a -valued function defined by

where.

By an abuse of notation, we use the same name for three functions with different domains.

Lemma 16. Let be a tangent cone. Then, is a height function with respect to defined in Definition 15.

Proof. It can be proved in the same way as the proof of Lemma 9.

Proposition 4. Let. Let be a tangent cone. Then, is constant on, i.e.,

Proof. Since, the result follows immediately.

Finally, let us consider briefly vector fields on a line segment space (Figure 6(a)) which are induced by a two-dimensional tangent cones. In the following section, vector fields of line segments will appear on the contour of a union of trajectories of triangles.

Definition 34 (The two-dimensional lattice). Let be the two-dimensional lattice generated by two vectors and, i.e.,

Definition 35 (The symmetric group on two letters). Let be the group of all the permutations of the set. consists of an identity element and.

Definition 36 (The set B₁ of all flat line segments). Let and. The slant line segment is defined by

Let be the set of all slant tetrahedrons, i.e.,

The shift operator on is defined by

Then, an equivalence relation is defined on by

The -equivalence class of is called a flat line segment and denoted by.

The set of all flat triangles is denoted by, i.e.,

Lemma 17. is a line segment space (Definition 3).

Proof. It follows immediately from the definition.

By an abuse of notation, the “image on” of a vertex v of is also denoted by. The tangent space at (Definition 5) is then given by

where. For simplicity, we often identify the vertex with. We shall then obtain a one-to-one correspondence

Definition 37 (Tangent bundle). The tangent bundle of (Definition 9) is given by

Definition 38 (Gradient). Let. The gradient of t is defined by

That is, is a -valued function on. The “vertex” is called the boundary vertex of. (Strictly speaking, is a set of one element. Here, we identify the set with its only element.)

Example 16. Shown in Figure 6(b) are all the four types of gradients of line segments (Definition 5). In the case of, only regular line segments are allowed.

Lemma 18. Let. Then,

That is, induces a -valued function on. By an abuse of notation, the induced function is also denoted by, i.e.,

Proof. It can be proved in the same way as the proof of Lemma 3.

Lemma 19. Let. Then, the local trajectory at associated with (Definition 7) is either

or

where. The local trajectory is called the local trajectory associated with.

Proof. Note that is the only vertex of that is not the boundary vertex. Because of the boundary vertex between and, local trajectories only go in one direction.

Two-dimensional “mountains of unit cubes” is defined as follows.

Definition 39 (A tangent cone Cone A). Let A be a finite subset of. A two-dimensional tangent cone is defined by

The set of all the slant line segments on the surface of is denoted by, i.e.,

The surface of a tangent cone induces a vector field of.

Definition 40 (Vector field on). Let be a tangent cone. The vector field on induced by c is defined by

is called a -embeddable vector field of line segments. Note that has no singular line segment.

Example 17. Shown in Figure 6(c) is a tangent cone

and the vector field on induced by the cone (top).

Local trajectories of on (Definition 11) is computed as follows.

Lemma 20. Let. Let be a tangent cone. Suppose that . Then, the local trajectory of at s is either

Proof. See Lemma 19.

Example 18. Let us consider a trajectory of the vector filed shown in Figure 6(c).

By connecting overlapping local trajectories, we shall obtain three maximal trajectories of line segments (Definition 12, 13), i.e.,

Proposition 5. Let V be a vector field on without singular line segments. Then,

Proof. It follows immediately from the definition.

Vector fields induced by a tangent cone are inherently associated with an U/D function and a height function.

Definition 41 (U/D function). Let. Let be a tangent cone. Let be the local trajectory of at s (i.e. or). The U/D function at s along the trajectory associated with is defined by

Lemma 21. Let be a tangent cone. Then, is an U/D function defined in Definition 14.

Proof. It follows immediately from the definition.

Example 19. Let us consider a flow of the vector filed shown in Figure 6(c).

Definition 42 (Height function). The height function on is a -valued function defined by

The height function on is a -valued function defined by

Let be a tangent cone. Then, the height function on associated with is a -valued function defined by

where.

By an abuse of notation, we use the same name for three functions with different domains.

Lemma 22. Let be a tangent cone. Then, is a height function with respect to defined in Definition 15.

Proof. It follows immediately from the definition.

Example 20. Let us consider a flow of the vector filed shown in Figure 6(c). Suppose that. Then,