Figure 2 is explained in this subsection. Shown in Figure 2(a) is the honeycomb mesh obtained by dividing a 2D Euclidean plane R² into a set of regular hexagons. H denotes the set of all hexagons of the mesh. A subset S of H is called connected if each h a ∈ S shares a side with another h b ∈ S (i.e., for each h a ∈ S , there exists another h b ∈ S such that h a and h b share a side).

Shown in Figure 2(b) is a connected subset S = { h 1 , h 2 , ⋯ , h 7 } of H. Since hexagons of H do not overlap each other, we write

S = h 1 ⊕ h 2 ⊕ ⋯ ⊕ h 7 . (1)

If S consists of only one hexagon h 1 , we write either S = h 1 or S = { h 1 } .

Shown in Figure 2(c) is an integral region on H, defined as follows. (Addition “+” of hexagons will be defined later in this subsection.)

Definition 2.1. (Integral Region) An integral region m 0 on H is a hole-free subset of R² covered by a connected finite subset S = { h 1 , h 2 , ⋯ , h n } of H. The hexagonal base of m 0 is then defined by

( m 0 ) H : = h 1 ⊕ h 2 ⊕ ⋯ ⊕ h n . (2)

For example, the hexagonal base of m 0 of Figure 2(c) is shown in Figure 2(b). I H denotes the set of all integral regions on H. Note that H ⊂ I H , i.e., hexagons of H are integral regions.

The set difference between two integral regions m 1 , m 2 ∈ I H is defined by

m 1 \ m 2 : = ∪ { h ∈ H | h ⊂ m 1   and   h ⊄ m 2 } , (3)

where ∪ { h 1 , h 2 , ⋯ , h n } : = h 1 ∪ h 2 ∪ ⋯ ∪ h n . The hexagonal base ( m 1 \ m 2 ) H of m 1 \ m 2 is defined in the same way as for integral regions (it may have holes).

Lemma 2.2. Let m 1 , m 2 ∈ I H . Then, m 1 \ m 2 ∈ I H if m 2 ⊄ m 1 .

Shown in Figure 2(d) are region-intermediates on H, defined as follows. Let M 0 = { m 1 , m 2 , ⋯ } ⊂ I H . M 0 is called connected if each m i ∈ M 0 shares a side with another m j ∈ M 0 . M 0 is called disjoint if m i ’s do not overlap each other. If M 0 is disjoint, we write

M 0 = m 1 ⊕ m 2 ⊕ ⋯ . (4)

If M 0 consists of only one integral region m 1 , we write either M 0 = m 1 or M 0 = { m 1 } .

Definition 2.3. (Region-Intermediate) A region-intermediate M 0 on H is a connected finite disjoint subset S = { m 1 , m 2 , ⋯ , m n } of I H . Since S is disjoint, we write

M 0 = m 1 ⊕ m 2 ⊕ ⋯ ⊕ m n . (5)

The hexagonal base of M 0 is then defined by

( M 0 ) H : = ( m 1 ) H ⊕ ( m 2 ) H ⊕ ⋯ ⊕ ( m n ) H . (6)

RI denotes the set of all region-intermediate on H.

Finally, addition of integral regions is defined using addition of directed polygonal chains as shown below.

Definition 2.4. (Directed Polygonal Chain) Let P 1 , P 2 , ⋯ , P n , P n + 1 be points in R². A directed polygonal chain P 1 P 2 ⋯ P n + 1 in R² is a set of directed line segments defined by

P 1 P 2 ⋯ P n + 1 : = { P 1 P 2 , P 2 P 3 , ⋯ , P n P n + 1 } , (7)

where P i P j denotes the directed line segment from P i to P j . If P n + 1 = P 1 , we obtain a closed directed polygonal chain in R². | P 1 P 2 ⋯ P n P 1 | denotes the area of R² bounded by P 1 P 2 ⋯ P n P 1 in R².

Let C 0 = { c 1 , c 2 , ⋯ } be a set of closed directed polygonal chains in R². C 0 is called disjoint if | c i | ’s do not overlap each other. If C 0 is disjoint, we write

C 0 = c 1 ⊕ c 2 ⊕ ⋯ . (8)

If C 0 is disjoint, the area | C 0 | of R² bounded by C 0 is defined by

| C 0 | : = | c 1 | ⊕ | c 2 | ⊕ ⋯ . (9)

Let m 0 ∈ I H . Since m 0 has no hole, we have

m 0 = | P 1 P 2 ⋯ P n P 1 | (10)

for some P 1 , P 2 , ⋯ , P n ∈ R 2 , where the vertices are labeled counter-clockwise. In the case of Figure 2(c),

m 0 = | P 1 P 2 ⋯ P 20 P 1 | . (11)

Definition 2.5. (The Boundary Operator ∂ on RI) Let m 0 = | P 1 P 2 ⋯ P n P 1 | ∈ I H , where the vertices are labeled counter-clockwise. The boundary ∂ m 0 of m 0 is defined by

∂ m 0 : = P 1 P 2 ⋯ P n P 1 . (12)

In this paper, the boundary of an integral region is always given the counter- clockwise orientation. Let M 0 = m 1 ⊕ m 2 ⊕ ⋯ ⊕ m n ∈ R I . Since { ∂ m 1 , ∂ m 2 , ⋯ , ∂ m n } is disjoint, the boundary ∂ M 0 of M 0 is defined by

∂ M 0 : = ∂ m 1 ⊕ ∂ m 2 ⊕ ⋯ ⊕ ∂ m n (13)

(See Equation (8)) Note that | ∂ M 0 | = M 0 .

Addition of integral regions is defined as follows.

Definition 2.6. ( m 1 + m 2 ) Let m 1 = | P 1 P 2 ⋯ P n P 1 | , m 2 = | Q 1 Q 2 ⋯ Q n Q 1 | ∈ I H such that they do not overlap. Addition of ∂ m 1 and ∂ m 2 is defined by

∂ m 1 + ∂ m 2 : = { P i P j ∈ ∂ m 1 | P j P i ∉ ∂ m 2 } ∪ { Q i Q j ∈ ∂ m 2 | Q j Q i ∉ ∂ m 1 } . (14)

In other words, the same line segments in opposite directions (i.e., P i P j and P j P i ) are cancelled when added. Addition of m 1 and m 2 is then defined by

m 1 + m 2 : = | ∂ m 1 + ∂ m 2 | . (15)

In the case of Figure 2(c), we have

m 0 = h 1 + h 2 + ⋯ + h 7 . (16)

Let’s denote the set of all natural numbers { 1 , 2 , 3 , ⋯ } by N .

Lemma 2.7.

The set of all integral regions on H is given by

I H = { h 1 + ⋯ + h n | n ∈ N , { h i } ⊂ H   isconnectedandhole-free } . (17)

The set of all region-intermediate on H is given by

R I = { m 1 ⊕ ⋯ ⊕ m n | n ∈ N , { m i } ⊂ I H   isconnected   and   disjoint } . (18)

Proof. They follow immediately from the definitions.∎

Remark 2.8. Integral domains play the role that “integers” do for rational numbers. That is, “rational” regions are obtained by dividing integral regions into loops of triangles [11] .

Figure 3 is explained in this subsection. Shown in Figure 3(a) is the triangular mesh obtained by dividing every hexagon of H into 6 equilateral triangles. B denotes the set of all triangles of the mesh.

Definition 2.9. (Trajectories of Triangles) A trajectory of triangles on B is a sequence of triangles of B connected by a common edge. (No direction is assigned to a trajectory.) The edges not used to connect adjacent triangles are called the normal edges (of the trajectory) at the triangle (i.e., the “normal vector” of the trajectory). In figures, normal edges are indicated by thick line segments.

Definition 2.10. (Loops of Triangles) A loop on B is a closed trajectory of triangles on B. In this paper, protein-like molecules are represented as a loop of triangles of B. | l p | denotes the area in R² swept by a loop l p , where the area enclosed by l p is also included in | l p | (for example, ϕ of Figure 1(a)).

A loop l p 0 of length 6 is called a hexagonal loop. A hexagonal loop l p 0 is denoted by h 0 if | l p 0 | = h 0 ∈ H . In other words, h 0 ∈ H denotes both a loop of length 6 and a hexagon of H, i.e., | h 0 | = h 0 .

Remark 2.11. The hexagonal base ( M 0 ) H of M 0 ∈ R I defined above is a region-intermediate consisting of hexagons as well as a loop-intermediate consisting of loops of length 6.

Shown in Figure 3(b) is a set L = { h 1 , h 2 , ⋯ , h 7 } of seven hexagonal loops on H. Since h i ’s do not overlap each other, we write

L = h 1 ⊕ h 2 ⊕ ⋯ ⊕ h 7 . (19)

If L consists of only one hexagon h 1 , we write either L = h 1 or L = { h 1 } .

Shown in Figure 3(c) is an integral loop on B, defined as follows.

Definition 2.12. (Integral Loop) A loop l p 0 on B is called integral if | l p 0 | is an integral region on H. I B denotes the set of all integral loops, i.e.,

I B = { l p 0 | l p 0   isaloopon   B   suchthat   | l p 0 | ∈ I H } . (20)

l p 0 ∈ I B is called an implementation of m 0 ∈ I H if | l p 0 | = m 0 . For example, l p 0 of Figure 3(c) is an implementation of m 0 of Figure 2(c).

Let L = { l p 1 , l p 2 , ⋯ } ⊂ I B . The set | L | of integral regions associated with L is defined by

| L | : = { | l p 1 | , | l p 2 | , ⋯ } ⊂ I H . (21)

If | L | is disjoint, we write

L = l p 1 ⊕ l p 2 ⊕ ⋯ . (22)

If L consists of only one integral loop l p 1 , we write either L = l p 1 or L = { l p 1 } .

Shown in Figure 3(d) top is a loop-intermediate on B, defined as follows.

Definition 2.13. (Loop-Intermediate) A loop-intermediate on B is a finite subset L 0 = { l p 1 , l p 2 , ⋯ , l p n } of I B such that | L 0 | is a region-intermediate on H. Since | L 0 | is disjoint, we write

L 0 = l p 1 ⊕ l p 2 ⊕ ⋯ ⊕ l p n . (23)

LI denotes the set of all loop-intermediate on B, i.e.,

L I : = { l p 1 ⊕ ⋯ ⊕ l p n | n ∈ N , l p i ∈ I B ( i = 1 , 2 , ⋯ , n ) , | l p 1 ⊕ ⋯ ⊕ l p n | ∈ I H } . (24)

Let M 0 ∈ R I . L 0 ∈ L I is called an implementation of M 0 if | L 0 | = M 0 . Note that some region-intermediates have no implementation. For example, m 3 of Figure 2(d) has no implementation (Figure 3(d) bottom).

Now, fusion and fission of integral loops are defined using addition of the corresponding integral regions (Addition of integral loops is considered in Section 4 below).

Definition 2.14. (Fusion and Fission of Integral Loops) Let l p 0 ∈ I B . Let L 0 = l p 1 ⊕ l p 2 ⊕ ⋯ ⊕ l p n ∈ L I . l p 0 is called the fusion of L 0 if

| l p 0 | = | l p 1 | + | l p 2 | + ⋯ + | l p n | . (25)

L 0 is then called a fission of l p 0 . In Figure 3, both h 1 ⊕ h 2 ⊕ ⋯ ⊕ h 7 and l p 1 ⊕ h 4 ⊕ p 2 are fissions of l p 0 .

Finally, let’s define flows of triangles on B.

Definition 2.15. (Flows of Triangles on B) A flow ψ of triangles on B is an assignment of normal edges to triangles of B, i.e.,

ψ ( t ) : = thesetofnormaledgesof   t     ( t ∈ B ) . (26)

ψ is called regular at t if ψ ( t ) consists of one edge (i.e., t is connected to exactly two adjacent triangles). ψ is called regular if ψ is regular at all triangles of B. F L W R denotes the set of all regular flows on B.

A triangle t of B is called singular if it is not regular. Singular triangles are called branch, terminal, or isolated triangles when they have no normal edges (i.e., connected to all the adjacent triangles), two normal edges (i.e., connected to only one adjacent triangle), or three normal edges (i.e., connected to no adjacent triangles), respectively.

Remark 2.16. We often consider trajectories of triangles without explicit reference to the corresponding flow ψ. A triangle t is then called regular if ψ is regular at t.

Definition 2.17. (Disjoint Unions L ( ψ ) and M ( ψ ) ) Let ψ ∈ F L W R . L ( ψ ) denotes the set of all the loops of ψ. Since trajectories of ψ do not overlap, we have

L ( ψ ) : = l p 1 ⊕ l p 2 ⊕ ⋯ , (27)

where l p i ’s are the loops of ψ. L ( ψ ) is called the loops associated with ψ. M ( ψ ) denotes the associated regions, i.e.,

M ( ψ ) : = | l p 1 | ⊕ | l p 2 | ⊕ ⋯ , (28)

M ( ψ ) is called the regions associated with ψ.

Lemma 2.18. Let ψ ∈ F L W R .

1) L ( ψ ) ∈ L I if L ( ψ ) is finite, connected, and hole-free.

2) M ( ψ ) ∈ R I if M ( ψ ) is finite and connected.

In the loop model of protein-like molecules, the shape of a new molecule is an integral region m 0 . A new molecule of the shape m 0 then is an implementation l p 0 of m 0 . The problem we consider in this paper is now defined as follows.

Problem 2.19. (Design of Protein-like Molecules) Given M 0 = { m 0 } ∈ R I , find L 0 = { l p 0 } ∈ L I such that | l p 0 | = m 0 .

In the next section, the problem is rephrased using a differential geometric structure on B.

Shown in Figure 4(a) is a unit cube in R³ and its orthogonal projection on the plane H₀ in R³ defined by

H 0 : = { ( x , y , z ) ∈ R 3 | x + y + z = 0 } . (29)

H is embedded in H₀ using unit cubes in R³, as explained below.

Definition 3.1. (Unit Cubes in R³) Let ( a , b , c ) ∈ R 3 . [ a , b , c ] denotes the unit cube at ( a , b , c ) , i.e.,

[ a , b , c ] : = [ a , a + 1 ] × [ b , b + 1 ] × [ c , c + 1 ] ⊂ R 3 , (30)

where [ x , y ] is the closed interval in R between x and y. If P = ( a , b , c ) ∈ R 3 , then [ a , b , c ] is also written as [ P ] . UC denotes the set of all unit cubes at the integer lattice Z 3 , i.e.,

U C : = { [ a , b , c ] | ( a , b , c ) ∈ Z 3 } . (31)

The height h t U C ( [ a , b , c ] ) of [ a , b , c ] ∈ U C is defined by

h t U C ( [ a , b , c ] ) : = a + b + c . (32)

Remark 3.2. [ a , b , c ] is given by

[ a , b , c ] = { ( a , b , c ) + ( u , 0 , 0 ) + ( 0 , v , 0 ) + ( 0 , 0 , w ) ∈ R 3 | ( u , v , w ) ∈ R 3                               suchthat   u + v + w = 1   and   u , v , w ≥ 0 } . (33)

Shown in Figure 4(a) top is a unit cube [ a , b , c ] ∈ U C with vertices O = ( a , b , c ) , P = ( a + 1 , b , c ) , Q = ( a , b + 1 , c ) , R = ( a , b , c + 1 ) , U = ( a + 1 , b + 1 , c ) , V = ( a , b + 1 , c + 1 ) , W = ( a + 1 , b , c + 1 ) , and X = ( a + 1 , b + 1 , c + 1 ) . The vertical diagonals OU, OV, and OW are drawn as thick line segments.

Definition 3.3. (Projection π of R³ onto H₀) π is the orthogonal projection of R³ onto H 0 defined by

π ( x , y , z ) : = ( ( 2 x − y − z ) / 3 , ( − x + 2 y − z ) / 3 , ( − x − y + 2 z ) / 3 ) . (34)

Shown in Figure 4(a) bottom is the orthogonal projection of OU, OV, and OW onto H 0 , forming part of a hexagonal mesh on H 0 .

Definition 3.4. (“Bumpy” Mesh H b u m p ) H b u m p is the “bumpy” honeycomb mesh defined on the top surfaces of

U C 0 : = { [ x , y , x ] ∈ U C | h t U C ( [ x , y , z ] ) = 0 } . (35)

The edges of the mesh are the vertical diagonals. H b u m p denotes the set of “bumpy” hexagons drawn on U C 0 (Figure 4(b) top).

Shown in Figure 4(b) bottom is the projection of H b u m p onto H 0 by π. In the following, we identify H with π ( H b u m p ) ⊂ H 0 . An embedding of B in H 0 is then obtained by dividing every hexagon of π ( H b u m p ) into 6 equilateral triangles.

Shown in Figure 4(c) top-left is a tangent cone C 0 to a region-intermediate M 0 , defined as follows. Roughly speaking, a 3D cone with multiple tops is obtained by stacking unit cubes diagonally (from ( ∞ , ∞ , ∞ ) to ( − ∞ , − ∞ , − ∞ ) ).

Definition 3.5. (Tangent Cone Cone A) Let A ⊂ Z 3 . The tangent cone C o n e A generated by A is defined by

C o n e A : = { ( x , y , z ) ∈ R 3 | max ( a , b , c ) ∈ A { min { x − a , y − b , z − c } } ≥ 0 } . (36)

TCONE denotes the set of all tangent cones, i.e.,

T C O N E : = { C o n e A | A ⊂ Z 3 } . (37)

P (TCONE) denotes the set of all subsets of TCONE, i.e., the power set of TCONE.

Definition 3.6. (Tops and Bottoms of Cone A) Let C 0 ∈ T C O N E . The top vertices of C 0 are the peaks of the cone. t o p ( C 0 ) denotes the set of all top vertices of C 0 . Note that C 0 = C o n e t o p ( C 0 ) . The bottom vertices of C 0 are the dents of the cone which are peaks if we look up the cone from ( ∞ , ∞ , ∞ ) . b o t t o m ( C 0 ) denotes the set of all bottom vertices of C 0 .

We define flows of triangles on B using tangent cones in R³.

Definition 3.7. (The Flow ψ C 0 on B) Let C 0 ∈ T C O N E . Note that the surfaces of C 0 consist of the top faces of unit cubes of UC. Taking their vertical diagonals as normal edge, we obtain a regular flow of triangles on the surface of C 0 (Figure 4(c) top-right). Projecting the regular flow onto H 0 by π, we obtain a regular flow on B. The regular flow is called the flow on B induced by C 0 and denoted by ψ C 0 .

Definition 3.8. F L W T C O N E denotes the set of all regular flows on B induced by tangent cones, i.e.,

F L W T C O N E : = { ψ C | C ∈ T C O N E } ⊂ F L W R . (38)

R I T C O N E and L I T C O N E denote the corresponding set of region-intermediates and loop-intermediates, respectively, i.e.,

R I T C O N E : = { M ∈ R I | ∃ C ∈ T C O N E suchthat M = M ( ψ C ) } . (39)

L I T C O N E : = { L ∈ L I | ∃ C ∈ T C O N E suchthat L = L ( ψ C ) } . (40)

The author has no proof of the following claim.

Claim 3.9. F L W T C O N E = F L W R .

In this paper, only flows of FLW_TCONE are considered.

Remark 3.10. R I T C O N E ≠ R I . For example, m 3 of Figure 2(d) is not contained in R I T C O N E .

Here we define a tangent “space” to M 0 ∈ R I .

Definition 3.11. (The Boundary Cone B C ( ∂ M 0 ) ) Let

M 0 = m 1 ⊕ m 2 ⊕ ⋯ ⊕ m n ∈ R I T C O N E , (41)

where

m i = | P i 1 P i 2 ⋯ P i k i P i 1 |     ( i = 1 , 2 , ⋯ , n ) . (42)

The boundary cone B C ( ∂ M 0 ) to ∂ M 0 is defined by

B C ( ∂ M 0 ) : = C o n e { Q i 1 , Q i 2 , ⋯ , Q i k i | i = 1 , 2 , ⋯ , n } , (43)

where Q i j ’s are points on the top surfaces of U C 0 such that

π ( Q i j ) = P i j . (44)

Remark 3.12. Since π is a one-to-one mapping between the top surfaces of U C 0 and H 0 , the boundary cone B C ( ∂ M 0 ) exists for all M 0 ∈ R I T C O N E .

Definition 3.13. (The Set TM₀ of Tangent Cones) Let M 0 ∈ R I T C O N E . The set T M 0 of tangent cones to M 0 is defined by

T M 0 : = { C ∈ T C O N E | t o p ( B C ( ∂ M 0 ) ) ∪ b o t t o m ( B C ( ∂ M 0 ) ) ⊂ s u r ( C ) } , (45)

where s u r ( C ) is the surfaces of C, i.e.,

s u r ( C ) : = { ( x , y , z ) ∈ R 3 | max ( a , b , c ) ∈ t o p ( C ) { min { x − a , y − b , z − c } } = 0 } . (46)

Note that B C ( ∂ M 0 ) ∈ T M 0 .

Remark 3.14. C ∈ T M 0 dose not imply t o p ( B C ( ∂ M 0 ) ) ⊂ t o p ( C ) .

Lemma 3.15. (The Base Tangent Cone C b a s e ( M 0 ) ) Let M 0 ∈ R I T C O N E . There exists C b a s e ( M 0 ) ∈ T M 0 such that

L ( ψ C b a s e ( M 0 ) ) = ( M 0 ) H , (47)

h t U C ( [ a , b , c ] ) = 0 for ( a , b , c ) ∈ t o p ( C b a s e ( M 0 ) ) . (48)

C b a s e ( M 0 ) is called the base tangent cone associated with M 0 .

Proof. Since π gives a one-to-one mapping between the top surfaces of U C 0 and H 0 , the result follows immediately.∎

Definition 3.16. (Mapping T) Assigning T M 0 to each M 0 ∈ R I T C O N E , we obtain a mapping T from R I T C O N E to P (TCONE). Let S ⊂ R I T C O N E . A section σ of T on S is a mapping from S to P (TCONE) such that M ( ψ σ ( M i ) ) = M i for all M i ∈ S . Γ T ( S ) denotes the set of all sections of T on S.

The design problem is now rephrased as follows.

Problem 3.17. (Design of Protein-like Molecules) Given M 0 = { m 0 } ∈ R I T C O N E , find C 0 ∈ T M 0 such that M ( ψ C 0 ) = M 0 .

Shown in Figure 5(a) is a binary relation over R I T C O N E , defined as follows.

Definition 4.1. (Binary Relation ≤ over R I T C O N E ) Let M a , M b ∈ R I T C O N E . Then, M a ≤ M b if and only if, for any m b ∈ M b ,

there exist m 1 , ⋯ , m n ∈ M a such that m b = m 1 + ⋯ + m n . (49)

In figures, we often use the arrow M a → M b to indicate M a ≤ M b .

Shown in Figure 5(b) is the binary relation over L I T C O N E induced by the binary relation ≤ over R I T C O N E . That is,

Definition 4.2. (Binary Relation ≤ over L I T C O N E ) Let L a , L b ∈ L I T C O N E . Then, L a ≤ L b if and only if, for any l p b ∈ L b ,

there exists l p 1 , ⋯ , l p n ∈ L a such that | l p b | = | l p 1 | + ⋯ + | l p n | . (50)

In figures, we often use the arrow L a → L b to indicate L a ≤ L b .

Remark 4.3. Notations such as ( R I T C O N E , ≤ ) and ( L I T C O N E , ≤ ) are used to explicitly indicate the binary relation equipped with a set.

Lemma 4.4. Let M 1 , M 2 ∈ R I T C O N E . Then,

T ( M 1 ) ⊂ T ( M 2 ) if M 1 ≤ M 2 . (51)

That is, T is a “covariant” mapping from ( R I T C O N E , ≤ ) to ( P ( T C O N E ) , ⊂ ) .

Shown in Figure 5(c) is examples of the greatest lower bound of loop-inter- mediates, defined as follows.

Definition 4.5. (⋀S and ⋁S) Let S ⊂ R I T C O N E . The greatest lower bound ⋀S of S is the greatest element of R I T C O N E that is less than or equal to each element

of S. The least upper bound ⋁S of S is the least element of RI that is greater than or equal to each element of S. ⋀S and ⋁S for S ⊂ L I T C O N E are also defined similarly.

Remark 4.6. In general, there are multiple candidates for ⋀S and ⋁S. In such cases, select one of them arbitrarily. Because of this uncertainty, “ M 0 ≤ M i for all M i ∈ S ” does not imply M₀ ≤ ⋀S.

Remark 4.7. H ≤ M 0 ≤ ∅ for any M 0 ∈ R I T C O N E , where ∅ denotes the empty set.

We use the following lemma to find “subsets” of a region-intermediate.

Lemma 4.8. Let M 0 ∈ R I T C O N E and C ∈ T C O N E . Then,

M ( ψ C ) ≤ M 0 If and only if C ∈ T M 0 . (52)

Proof. M ( ψ C ) ≤ M 0 if and only if ∂ M 0 ⊂ ∂ M ( ψ C ) . ∂ M 0 ⊂ ∂ M ( ψ C ) if and only if t o p ( B C ( ∂ M 0 ) ) ∪ b o t t o m ( B C ( ∂ M 0 ) ) ⊂ s u r ( C ) . The result follows immediately. ∎

Two types of coverings are defined as follows.

Definition 4.9. (Coverings of a Region-Intermediate) Let M 0 ∈ R I T C O N E . Let S ⊂ R I T C O N E . S is called a covering of M₀ if ⋁S = M₀.

Definition 4.10. (Topological Coverings of an Integral Region) Let m 0 ∈ I H . Let V = { c 1 , c 2 , ⋯ , c n } ⊂ I H . V is called a topological covering of m 0 if 1) m 0 = ∪ i = 1 n c i , and 2) for each c i ∈ V , there exists another c j ∈ V such that c i ∩ c j ≠ ∅ .

Remark 4.11. Since some integral regions have no implementation (i.e., there exists c i ∈ I H such that c i ≠ | l p | for any l p ∈ I B ), topological coverings may have no sections on them.

Lemma 4.12. Let m 0 ∈ I H . Let { c 1 , c 2 , ⋯ , c n } ⊂ I H be a topological covering of . A covering of m 0 is then obtained by

{ m 0 ∧ c 1 , m 0 ∧ c 2 , ⋯ , m 0 ∧ c n } . (53)

Example 4.13. In Figure 5(c), { A 1 , B 1 } is a topological covering of X. { A 1 ∧ X , B 1 ∧ X } is a covering of X.

The proposed design method uses a specific type of covering (in Problem 4.38).

Definition 4.14. (Hexagonal Covering S_V of an Integral Region) Let V = { c 1 , c 2 , ⋯ , c n } ⊂ I H be a topological covering of m 0 ∈ I H . The hexagonal covering S V of m 0 associated with V is defined by

S V = { c 1 ⊕ ( m 0 \ c 1 ) H , c 2 ⊕ ( m 0 \ c 2 ) H , ⋯ , c n ⊕ ( m 0 \ c n ) H } . (54)

Lemma 4.15. S V is a covering of m 0 .

The design problem is now rephrased as follows.

Problem4.16. (Incremental Design of Protein-like Molecules) Given 1) a target shape m 0 : M 0 = { m 0 } ∈ R I T C O N E , 2) a topological covering V of m 0 : V = { c 1 , c 2 , ⋯ , c n } , 3) a section σ of T on V: σ ∈ Γ T ( V ) . Then, compute C 0 ∈ T M 0 such that M ( ψ C 0 ) = M 0 by patching “local” loop-intermediates L ( ψ σ ( c 1 ) ) , L ( ψ σ ( c 2 ) ) , ⋯ , and L ( ψ σ ( c n ) ) together.

To patch loop-intermediates together, we define addition of loop-intermediates using transformations on TCONE, defined as follows.

Definition 4.17. (TRANS (TCONE)) A transformation on TCONE is a mapping from TCONE to TCONE. TRANS (TCONE) denotes the set of all transformations on .

Let A ∈ T R A N S ( T C O N E ) and C ∈ T C O N E . We use the symbol “ ⋅ ” to denote the transformation of C by A, i.e., A ⋅ C . A ⋅ C is also called the action of A on C. Let A 1 , A 2 , ⋯ , A n ∈ T R A N S ( T C O N E ) . We use the symbol “ ∘ ” to denote the composition of transformations, i.e.,

A 1 ∘ A 2 ∘ ⋯ ∘ A n ⋅ C : = A 1 ⋅ ( A 2 ⋅ ( ⋯ ⋅ ( A n ⋅ C ) ) ⋯ ) . (55)

Example 4.18. Unit cubes induce transformations on TCONE as follows. Let C ∈ T C O N E , where t o p ( C ) = { P 1 , P 2 , ⋯ , P n } , i.e.,

C = C o n e { P 1 , P 2 , ⋯ , P n } . (56)

Taking the unit cube [ a , b , c ] at P 1 = ( a , b , c ) from C, we obtain another tangent cone

C ′ = C o n e { P ′ 1 , P ′ 2 , P ′ 3 , P 2 , ⋯ , P n } , (57)

where P ′ 1 = ( a + 1 , b , c ) , P ′ 2 = ( a , b + 1 , c ) , and P ′ 3 = ( a , b , c + 1 ) .Conversely, putting the unit cube [ a , b , c ] on C ′ , we obtain the original cone C.

Definition 4.19. (The minimal L-cone C_L) Let L ∈ L I T C O N E and C ∈ T | L | . C is called a L-cone if L ( ψ C ) = L . The tangent cone C L is the minimal L-cone with respect to set inclusion, i.e., C L ⊂ C for any L-cone C. Since L ∈ L I T C O N E , C L ∈ T | L | always exists and uniquely determined by L.

Transformations on TCONE induce transformations on L I T C O N E as follows.

Definition 4.20. (Transformations on L I T C O N E ) Let A ∈ T R A N S ( T C O N E ) and L ∈ L I T C O N E . The transformation of L by A is defined by

A ⋅ L : = L ( ψ A ⋅ C L ) . (58)

A ⋅ L is called the action of A on L.

Definition 4.21. (Transformations P [ a , b , c ] , T [ a , b , c ] , and P T [ a , b , c ] ) Let [ a , b , c ] ∈ U C . Two transformations P [ a , b , c ] and T [ a , b , c ] on TCONE induced by [ a , b , c ] is defined by

P [ a , b , c ] ⋅ C : = C ∪ C o n e { ( a , b , c ) } , (59)

T [ a , b , c ] ⋅ C : = { [ x , y , z ] ∈ C | [ a , b , c ] ∉ C o n e { ( x , y , z ) } } , (60)

where C ∈ T C O N E . P [ a , b , c ] ⋅ C is called the put & fill-action by [ a , b , c ] on C. T [ a , b , c ] ⋅ C is called the take & clear-action by [ a , b , c ] on C. We denote the composition of P after T by PT, i.e.,

P T [ a , b , c ] : = P [ a , b , c ] ∘ T [ a , b , c ]     ( [ a , b , c ] ∈ U C ) . (61)

P T [ a , b , c ] ⋅ C is called the take & put-action by [ a , b , c ] on C.

Remark 4.22. After the action of P T [ a , b , c ] on C ∈ T C O N E , the cube [ a , b , c ] is always visible from ( − ∞ , − ∞ , − ∞ ) . On the other hand, after the action of P [ a , b , c ] on C, [ a , b , c ] may not be visible from ( − ∞ , − ∞ , − ∞ ) .

Lemma 4.23. Let [ a , b , c ] ∈ U C and C ∈ T C O N E . Then,

P [ a , b , c ] ⋅ C = C if [ a , b , c ] ∈ C , (62)

T [ a , b , c ] ⋅ C = C if [ a , b , c ] ∉ C . (63)

Definition 4.24. ( T R A N S 〈 T , P T 〉 ( T C O N E ) ) T R A N S 〈 T , P T 〉 ( T C O N E ) denotes the set of all the transformations on TCONE generated by finite compositions of T [ a , b , c ] and P T [ a , b , c ] ( [ a , b , c ] ∈ U C ), i.e.,

T R A N S 〈 T , P T 〉 ( T C O N E ) : = { A 1 ∘ ⋯ ∘ A n | n ∈ Z , A i = P u   or   P T u   forsome   u ∈ U C } . (64)

In general, G 1 ∘ G 2 ⋅ C ≠ G 2 ∘ G 1 ⋅ C for G 1 , G 2 ∈ T R A N S 〈 T , P T 〉 ( T C O N E ) and C ∈ T C O N E

Example 4.25. Let u 1 = [ a , b , c ] , u 2 = [ a ′ , b ′ , c ′ ] ∈ U C such that ( a , b , c ) ∈ C o n e { a ′ , b ′ , c ′ } . Then,

P T u 1 ∘ P T u 2 ⋅ C o n e { ( a ′ , b ′ , c ′ ) } ≠ P T u 2 ∘ P T u 1 ⋅ C o n e { ( a ′ , b ′ , c ′ ) } , (65)

T u 1 ∘ P T u 2 ⋅ C o n e { ( a ′ , b ′ , c ′ ) } ≠ P T u 2 ∘ T u 1 ⋅ C o n e { ( a ′ , b ′ , c ′ ) } . (66)

Definition 4.26 (Well-defined Transformations) Let G = A 1 ∘ A 2 ∘ ⋯ ∘ A n ∈ T R A N S 〈 T , P T 〉 ( T C O N E ) . G is called well-defined if the action of G on TCONE does not depend on the order of A i ’s, i.e.,

G ⋅ C = A ρ ( 1 ) ∘ A ρ ( 2 ) ∘ ⋯ ∘ A ρ ( n ) ⋅ C for all C ∈ T C O N E . (67)

for any permutation ρ of { 1 , 2 , ⋯ , n } .

Remark 4.27. If G is well-defined, removed unit cubes are removed forever and placed unit cubes are placed forever.

Addition of loop-intermediates is now defined using transformations on TCONE.

Definition 4.28. {Transformations on TM₀} Let M 0 ∈ R I . The set T R A N S ( T M 0 ) of transformations on T M 0 is defined by

T R A N S ( T M 0 ) : = { G ∈ T R A N S 〈 T , P T 〉 ( T C O N E ) | G ⋅ C ∈ T M 0   forall   C ∈ T M 0 } . (68)

Lemma 4.29. Let M 0 ∈ R I T C O N E and G 1 , G 2 ∈ T R A N S ( T M 0 ) . Then,

G 1 ∘ G 2 ∈ T R A N S ( T M 0 ) . (69)

Lemma 4.30. Let L 0 , L ∈ L I T C O N E and G ∈ T R A N S ( T | L 0 | ) . Then,

If L ≤ L 0 , then G ⋅ L ≤ L 0 . (70)

Definition 4.31. (The Relative Transformation G H ( L ) ) Let L ∈ L I T C O N E . The relative transformation G H ( L ) of L with respect to H is defined by

G H ( L ) : = P T [ P 1 ] ∘ ⋯ ∘ P T [ P n ] ∘ T [ Q 1 − ( 1 , 1 , 1 ) ] ∘ ⋯ ∘ T [ Q k − ( 1 , 1 , 1 ) ] , (71)

where

{ P 1 , ⋯ , P n } = { P ∈ t o p ( C L ) | h t U C ( [ P ] ) ≠ 0 } , (72)

{ Q 1 , ⋯ , Q k } = { Q ∈ b o t t o m ( C L ) | h t U C ( [ Q ] ) ≠ 2 } . (73)

[ Q − ( 1 , 1 , 1 ) ] is defined by [ Q − ( 1 , 1 , 1 ) ] : = ( a − 1 , b − 1 , c − 1 ) for Q = ( a , b , c ) .

Lemma 4.32. Let L ∈ L I T C O N E and M ∈ R I T C O N E such that | L | ≤ M . Then, G H ( L ) is well-defined and

L = G H ( L ) ⋅ L H ≤ G H ( L ) ⋅ M H . (74)

Remark 4.33. The hexagonal base M H is a loop-intermediate consisting of loops of length 6 as well as a region-intermediate consisting of hexagons.

Lemma 4.34. Let L ∈ L I T C O N E and M ∈ R I T C O N E such that | L | ≤ M . Then,

G H ( L ) ∈ T R A N S ( T M ) . (75)

Addition of loop-intermediates is now defined as follows.

Definition 4.35. (Addition of Loop-Intermediates) Let L 1 , L 2 , ⋯ , L n ∈ L I T C O N E and M ∈ R I T C O N E such that | L 1 | , | L 2 | , ⋯ , | L n | ≤ M . Then,

L 1 + L 2 + ⋯ + L n : = G H ( L 1 ) ∘ G H ( L 2 ) ∘ ⋯ ∘ G H ( L n ) ⋅ M H . (76)

Remark 4.36. Addition L 1 + L 2 + ⋯ + L n is defined with respect to M H , which is not explicitly indicated in the formula.

Definition 4.37. (Section σ S V on S V ) Let V = { c 1 , c 2 , ⋯ , c n } ⊂ I H be a topological covering of m 0 ∈ I H . Let σ ∈ Γ T ( V ) . The section σ S V of T on the hexagonal covering S V is defined by

σ S V ( c i ⊕ ( m 0 \ c i ) H ) : = σ ( c i ) ∪ C b a s e ( ( m 0 \ c i ) H ) . (77)

Note that

L ( ψ σ S V ( c i ⊕ ( m 0 \ c i ) H ) ) = L ( ψ σ ( c i ) ) ⊕ ( m 0 \ c i ) H ( i = 1 , ⋯ , n ) . (78)

Since c i ⊕ ( m 0 \ c i ) H ≤ m 0 ( i = 1 , ⋯ , n ) , we can define addition

∑ i = 1 n L ( ψ σ S V ( c i ⊕ ( m 0 \ c i ) H ) ) (79)

by Definition 4.35.

Using addition of loop-intermediates, the design problem is now rephrased as follows.

Problem 4.38. (Incremental Design of Protein-like Molecules) Given 1) a target shape m 0 : M 0 = { m 0 } ∈ R I T C O N E ; 2) a topological covering V of m 0 : V = { c 1 , c 2 , ⋯ , c n } ; 3) a section σ of T on V: σ ∈ Γ T ( V ) . Then, we obtain L 0 ∈ L I T C O N E such that | L 0 | ≤ M 0 by

L 0 : = ∑ i = 1 n L ( ψ σ S V ( c i ⊕ ( m 0 \ c i ) H ) )             = G H ( L 1 ) ∘ G H ( L 2 ) ∘ ⋯ ∘ G H ( L n ) ⋅ ( M 0 ) H , (80)

where L i : = L ( ψ σ S V ( c i ⊕ ( m 0 \ c i ) H ) ) . The question here is “when dose L₀ consist of a single loop?”

Figure 6 shows the effect of the action of T R A N S ( T M 0 ) on L ( ψ C b a s e ( M 0 ) ) using the height of normal edges, defined as follows.

Definition 5.2. (Height of Normal Edges) Let P 1 P 2 be a vertical diagonal of a top face of [ a , b , c ] ∈ U C . The height h t E ( P 1 P 2 ) of P 1 P 2 is defined by

h t E ( P 1 P 2 ) : = h t U C ( [ a , b , c ] ) . (82)

Let C ∈ T C O N E . Let Q 1 Q 2 be a normal edge of the induced flow ψ C such that π ( P 1 P 2 ) = Q 1 Q 2 , where P 1 P 2 is the corresponding vertical diagonal on the surfaces of C. The height h t E ( Q 1 Q 2 ) of Q 1 Q 2 (with respect to C) is defined by

h t E ( Q 1 Q 2 ) : = h t E ( P 1 P 2 ) . (83)

Shown in Figure 6(a) top is the top view of the base tangent cone

C b a s e ( M 0 ) ∈ T M 0 (84)

of some M 0 ∈ R I . Shown in Figure 6(a) bottom is the loop-intermediate

L ( ψ C b a s e ( M 0 ) ) (85)

on B. By definition, the heights of all normal edges of ψ C b a s e ( M 0 ) are 0.

Shown in Figure 6(b) top is the tangent cone

P T [ P 1 ] ∘ P T [ P 2 ] ∘ T [ P 3 ] ⋅ C b a s e ( M 0 ) . (86)

In the upper part, two Y-shaped sets of normal edges are replaced by two inverted Y-shaped sets of normal edges (thick line segments) by putting the two unit cubes [ P 1 ] and [ P 2 ] . In the lower part, a Y-shaped set of normal edges is replaced byan inverted Y-shaped set of normal edges (thick line segments) by taking the unit cube [ P 3 ] .

Shown in Figure 6(b) bottom is the loop-intermediate

P T [ P 1 ] ∘ P T [ P 2 ] ∘ T [ P 3 ] ⋅ L ( ψ C b a s e ( M 0 ) ) . (87)

The light grey area indicates the projection image of [ P 1 ] and [ P 2 ] by π. The dark grey area indicates the projection image of the removed [ P 3 ] by π. Note that normal edges of different heights are not directly connected.

Lemma 5.3. If two normal edges n 1 and n 2 of a flow of triangles are connected, the difference of their heights is even, i.e.,

h t E ( n 1 ) ≡ h t E ( n 2 ) mod 2 . (88)

In Figure 6(c) top, the normal edges of height −1 are connected to the normal edges of height 1 by putting the unit cube [ P 4 ] of height −1 on the tangent cone of Figure 6(b). In Figure 6(c) bottom, the light grey area (height −1) and the dark grey area (height 1) are now in contact.

In Figure 6(d) top, the normal edges of height −2 are connected to the normal edges of height 0 by putting the unit cube [ P 5 ] of height −2 on the tangent cone of Figure 6(c). In Figure 6(d) bottom, the white area indicates the projection image of [ P 5 ] . Note that the grey area (height 0) and the white area (height −2) are in contact.

In the loop model, it is easier to design an integral loop from scratch than to design a “hybrid” of known integral loops, since the area enclosed by a loop l p is included in | l p | . (In protein science, it is a formidable task to design a novel artificial protein from scratch.)

Let M 0 = { m 0 } ∈ R I T C O N E . Disconnecting normal edges of L ( ψ C b a s e ( M 0 ) ) along the boundary, we obtain l p ∈ I B such that | l p 0 | = m 0 as explained below.

Shown in Figure 7(a) top is the top view of the base tangent cone

C b a s e ( M 0 ) ∈ T M 0 (89)

of some M 0 ∈ R I T C O N E . Shown in Figure 7(a) bottom is the loop-intermediate

L ( ψ C b a s e ( M 0 ) ) (90)

on B. By definition, the heights of all normal edges of ψ C b a s e ( M 0 ) are 0.

In Figure 7(b), normal edges of height 0 is disconnected by putting unit cubes of height −1 along the boundary on C b a s e ( M 0 ) . The light grey area in Figure 7(b) bottom indicates the projection image of the added unit cubes.

In Figure 7(c), normal edges of height 0 is disconnected by taking unit cubes of height 0 along the boundary from C b a s e ( M 0 ) . The dark grey area in Figure 7(c) bottom indicates the projection image of the removed unit cubes.

In Figure 7(d) left, a loop-intermediate consisting of three integral loops is obtained by putting 8 unit cubes of height −1 and taking a unit cube of height 0 along the boundary. Then, taking another unit cube of height 0 at the meeting point of the boundaries of the three loops, we obtain the integral loop shown in Figure 7(d) right.

Sufficient conditions for L₀ of Problem 5.1 to be a loop are given using the two concepts defined below.

Definition 5.4. (The set N h t = 0 ( L ) of normal edges) Let L ∈ L I T C O N E . N h t = 0 ( L ) denotes the set of all the normal edges of height 0 contained in L.

Definition 5.5. (Rifts of a Loop-Intermediate) A crack of L ∈ L I T C O N E is a polygonal chain of normal edges of L connected to the boundary of L. A crack is called a rift if it consists of more than one normal edge.

Lemma 5.6. Let M 0 ∈ R I . Let L 1 , L 2 ∈ L I T C O N E such that | L 1 | , | L 2 | ≤ M 0 . Let σ ∈ Γ T ( { | L 1 | , | L 2 | } ) . If G H ( L 1 ) ∘ G H ( L 2 ) is well-defined, then

N h t = 0 ( L ( ψ σ ( | L 1 | ) ) + L ( ψ σ ( | L 2 | ) ) ) = N h t = 0 ( L ( ψ σ ( | L 1 | ) ) ) ∩ N h t = 0 ( L ( ψ σ ( | L 2 | ) ) ) . (91)

In general, the set N h t = 0 ( ∑ i L ( ψ σ ( | L i | ) ) ) shrinks monotonically as more loop- intermediates are added.

Proof. Because of Remark 4.27, removed normal edges of height 0 are removed forever. ∎

Proposition 5.7. (Sufficient Conditions to be a Loop) Settings are the same as for Problem 5.1. Let l p i = L ( ψ σ ( c i ) ) ∈ I B ( i = 1 , 2 ) , i.e.,

L ( ψ σ S V ( c i ⊕ ( m 0 \ c i ) H ) ) = l p i ⊕ ( m 0 \ c i ) H     ( i = 1 , 2 ) . (92)

Then, L 0 consists of a single loop if G H ( L 1 ) ∘ G H ( L 2 ) is well-defined and one of the following three conditions are satisfied:

1) No cracks of l p 1 and l p 2 connect to ∂ M 0 ∩ ∂ ( c 1 ∩ c 2 ) . There is at most only one rift of l p 1 that penetrates into c 1 ∩ c 2 through c 1 \ c 2 . No rift of l p 2 penetrates into c 1 ∩ c 2 through c 2 \ c 1 (Figure 8(a) and Figure 8(b)).

2) No cracks of l p 1 and l p 2 connect to ∂ M 0 ∩ ∂ ( c 1 ∩ c 2 ) . There is at most only one rift of l p 2 that penetrates into c 1 ∩ c 2 through c 2 \ c 1 . No rift of l p 1 penetrates into c 1 ∩ c 2 through c 1 \ c 2 .

3) Both l p 1 and l p 2 have a crack connected to ∂ M 0 ∩ ∂ ( c 1 ∩ c 2 ) . No rift of l p 1 penetrates into c 1 ∩ c 2 through c 1 \ c 2 . No rift of l p 2 penetrates into c 1 ∩ c 2 through c 2 \ c 1 (Figure 8(c)).

Proof. Since G H ( L 1 ) ∘ G H ( L 2 ) is well-defined, normal edges of height 0 are not added as a result of addition by Lemma 5.6. That is, cracks are extended only by normal edges of height n, where n ∈ N such that n ≠ 0 and n is even. “No cracks of l p 1 and l p 2 connect to ∂ M 0 ∩ ∂ ( c 1 ∩ c 2 ) ” implies M 0 is not separated by a polygonal chain contained in c 1 ∩ c 2 .

Since l p 1 and l p 2 bring no normal edges of height n ( n ≠ 0 ) into c 2 \ c 1 and c 1 \ c 2 , respectively, the result follows. ∎

Shown in Figure 9 are examples where the sufficient conditions are not satisfied.

In Figure 9(a), both l p 1 and l p 2 have a rift that penetrates into c 1 ∩ c 2 through c 1 \ c 2 and c 2 \ c 1 , respectively. In Figure 9(b), l p 1 has two rifts that penetrate into c 1 ∩ c 2 through c 1 \ c 2 . In Figure 9(c), both l p 1 and l p 2 have a crack connected to ∂ M 0 ∩ ∂ ( c 1 ∩ c 2 ) , and l p 1 has a rift that penetrates into c 1 ∩ c 2 through c 1 \ c 2 .

Figure 10(a) is the case given in Figure 1(a). Shown in Figure 10(a) bottom are all normal edges of height 0, where both l p 1 and l p 2 have a rift consisting

normal edges of height 0. All other normal edges are height 1. As a result of addition, some of the normal edges of height 0 are removed and we obtain the loop l p 0 .

Figure 10(b) is the case given in Figure 1(b). Shown in Figure 10(b) bottom are all normal edges of height 0 and height −1 (thick line segments). All other normal edges are height 1. In this case, G ( L 1 , M 0 ) ∘ G ( L 2 , M 0 ) is not well- defined and we cannot use Lemma 5.6. As a result of addition, the rift of l p ′ 2 is extended by normal edges of height 0 and we obtain two loops l p ′ 3 and l p ′ 4 .