# MODULE CONNECTION ARRANGEMENT PATTERNS OF A CELL

A cell can be composed of a varying number modules, sometimes in various combinations for a given number of modules. When there can be various combinations for a given number of modules, there can be some arrangements that are not appropriate. Figure 19a shows 3 Core Base Assemblies, the objects consisting geometrically of two rectangular boxes and a square truncated pyramid with sides that slope at 45° angles that are part of the modules, connected together in such a way that they share a common corner. One of these modules has a gray shaded truncated pyramid for future reference. In this case the arrangement of modules is appropriate.

Figure 19b also shows 3 Core Base Assemblies connected together in such a way that they share a common corner, but they are a mirror image pattern of the 3 Core Base Assemblies of Figure 19a. This is also an appropriate arrangement of modules. The Core Base Assemblies of Figures 19a and 19b are opposite halves of a cell that can be joined together to form a whole cell.

Figure 19c shows two modules connected together. This pattern is the only way any two modules can be joined together when there are only two; the bottom one can, for example, be assigned identification number 1 (ID #1) and the other would be assigned ID #2. There are four sides and two types of sides; the type of side is used as a reference to determine which has ID #1 or ID #2. If a third module is added to the pattern, using the Cube Face Numbering Convention article as a convention reference, then to Figure 19c the module with the gray pyramid in Figure 19b would be assigned ID #3; and in Figure 19a it would be assigned ID #4 since the first two modules are in reverse positions. Three of the arrangements of modules in Figure 19c can be joined together to form a whole cell.

Figure 19d shows the same two modules of Figure 19c in a 2-dimensional side view.

Figure 19e shows a third module added to the arrangement of modules from Figure 19d. This is not an appropriate arrangement of modules, and can be determined because there are more than two modules connected together and none of them share a common corner. One reason it is not appropriate is because if the pins-and-hole system that joins two modules together are not designed to latch the modules together (e.g., the pin is a simple cylinder), then this arrangement will not allow the modules to latch together to bond.

Figure 19f shows the arrangement of Figure 19e with a fourth module added to the back, shown partially out of view, in green, and with a gray pyramid. This arrangement of modules is appropriate, since each module shares a common corner with two of its adjacent modules. The arrangement of modules in Figure 19c can be joined to the arrangement of modules in Figure 19f to form a whole cell.

Figure 19g shows three modules arranged in the complementary arrangement of the ones in Figure 19e, and is also not an appropriate arrangement of modules for the same reason, and also because the two opposite modules won’t have the direct electronic connections described in the Pattern for Element Core Electronic Connections article. The two arrangements of modules in Figures 19e and 19g can join together to form a whole cell.

Figure 19h shows four modules in a ring pattern, and it is also not an appropriate arrangement of modules for the same reasons given for the three of Figure 19g.

Although a fourth module with a gray pyramid is added, shown partially out of view and in green and yellow, to the arrangement of Figure 19g to form the arrangement of Figure 19i, there is no difference in the pattern arrangements of Figures 19i and 19f other than the gray one being in a different position in the pattern.

Figure 19j shows five modules connected together, made by connecting a fifth module to the top of the pattern of modules shown in figure 19i. This is the only way five modules can be connected together to form part of a cell.

In Summary, the following table shows the possible number of arrangements for a given number of modules in a cell:

number of modules in a cell 2 3 4 5 6
possible number of arrangements 1 4 2 1 1

There are only two number-of-modules-in-a-cell situations where there are multiple possible number-of-arrangements of modules, for 3 and 4 modules. For 3 modules, 2 are appropriate and 2 are inappropriate; for 4 modules, 1 is appropriate and 1 is inappropriate.

# CONNECTING PLATES ALIGNMENT REQUIREMENT

When the connecting plates of two different cells engage each other to connect to bond together correctly, the axis going though the center of each telescoping shaft and connecting plate have to be both parallel and aligned with each other; in other words, they both have to share a common axis. The reason this is necessary is to be able to utilize the entire area of the connecting plate of each module against moment forces, and large tension or compression forces along the telescoping shaft (module leg) axes. Figure 17a shows part of two telescoping shafts and side views of their connecting plates for two different modules; one of the connecting plates is blue and the other is red. The dashed green line through each telescoping shaft and connecting plate indicates the axis of the leg and connecting plate, and at the same time shows that they both share a common axis.

In figure 17b, the two legs and connecting plates of figure 17a are shown as parallel but not aligned to each other to share a common axis.

If the two connecting plates from figure 17b try to engage to bond together along their axes, they will come in contact as shown in figure 17c; since they aren’t aligned they won’t be able to bond, at least not properly if even possible at all. This type of contact is acceptable if it is not necessary for the two modules to bond together and if there isn’t going to be an excessive amount of force or pressure between them.

Figure 17d shows the two modules of figure 17a in contact for correctly connecting together; they might or might not be bonded together.

# LATERAL LATTICE RECONFIGURATION USING LEG EXTENDING CAPABILITIES

This type of modular self-reconfigurable system is capable of lattice based transformations to change its shape by using the leg extending and retracting feature. Figure 6a shows several modules initially in a state where there is an empty space above module E and between modules B and C. The following example shows how the modules in this matrix can translate a module to fill in this empty space by moving another module to that location.

Figure 6b shows the first transition state of the matrix given in Figure 6a. It involves disconnecting and shortening the legs between modules B and D, and beginning the process of extending the legs between modules B and C to join them together.

Figure 6c shows module B translating horizontally by extending the leg connections between modules A and B until the extending legs between modules B and C join together.

Figure 6d shows modules A and B disconnecting from each other and retracting, and the use of module C to align module B above module E by shortening the legs connecting modules B and C together.

Figure 6e shows the legs for modules B and E extending and joining together and the legs that disconnected between modules A and B completely retracted; this completes the transition.

This matrix could also have accomplished the same goal, also with lattice transitions, by first extending the legs between modules B and C to connect to each other, then disconnecting and shortening the legs that join modules B and D together, then shortening the legs that join B and C together, then finally extending the legs between modules B and E to connect these modules to each other. This version describes how it is possible to accomplish this task without the first column of modules in the matrix.

# DIAGONAL LATTICE RECONFIGURATION USING LEG EXTENDING CAPABILITIES

This type of modular self-reconfigurable system is capable of lattice based transformations to change its shape by using the leg extending and retracting feature. Figure 7a shows a matrix consisting of a column on the left with 2 modules which is connected to another module in the middle column on the bottom row, and it is connected to the bottom of a column of 3 modules on the right. The following example shows how the modules in this matrix can translate a module to move it both horizontally and vertically.

Figure 7b shows the first transition state of the matrix given in Figure 7a. It involves extending the legs between modules A and B to translate module A vertically.

Figure 7c shows modules A and C extending legs between them and joining them together.

Figure 7d shows modules A and B disconnecting from each other and retracting the legs between them.

Figure 7e shows the legs between modules A and C retracting to translate module A horizontally.

Module A has now been translated out of the first column into the second column and out of the second row into the first row, and this was accomplished using only lattice transitions. In a similar matrix arrangement, the same type of goal could have been accomplished by first extending the legs of the two middle row modules to join, then disconnecting the legs between modules A and B, then shortening the legs that join module A to the module in the column of modules on the right to translate module A to the second column, then connecting the leg from module A to the module in the middle column and bottom row, then disconnecting the legs that join module A to the module in the column on the right, then extending the legs that join module A to the module in the bottom row to translate module A to the top row, then connecting the legs between modules A and C, then finally disconnecting and retracting the legs between module A and the module in the bottom row.

# ALIGNING LEGS FOR JOINING CONNECTING PLATES

To join together two modules that are not aligned, in positions where all that is necessary is to extend the appropriate legs, both of these legs have to pivot to the correct angle. Figure 9a, shown below, is an example of two modules A and B that are not connected directly to each other in a matrix.

Suppose the right leg of module A and top leg of module B are selected for connecting these modules together. Figure 9b, shown below, shows the common axis along which these legs must align for connection as a green line.

Figure 9c, shown below, shows the legs for modules A and B in new positions after they have pivoted to the angle to align them to their common axis, shown as a green line, for joining together.

These legs are now aligned so that all that is is necessary for modules A and D to join together is to extend the legs. Figure 9d, shown below, shows modules A and B joined together after the legs have been extended.

If modules A and B are in the same matrix system and approximal, then it is not necessary to use any sensors or external input data to determine that the legs are aligned for joining. If modules A and B are in different matrix systems or a significant enough distance apart to be able to geometrically introduce an alignment discrepancy in the same matrix system, then it is necessary to use sensors or external input data to determine that the legs are aligned for joining; this can be accomplished by placing several sensors on the surface of the connecting plate and a significant distance from the axis of the leg and connecting plate, near the perimeter of the connecting plate circle. Once they are in close enough proximity to detect the presence of each other by emitting signature signals, the differences in strength of the signals determines whether or not the legs need to pivot until the signal strengths are equal to each other. This is usually a basic “reflex” action; an exception would be when the legs reach their pivoting limit. This pivoting limit is a minimum of 45° in any direction from the center axis perpendicular to the face of the module core assembly, or core “cube,” and will probably not be more than a few degrees beyond this because of the limitations imposed by the design of the module’s mechanical structures. In a situation where the legs cannot pivot to align to their common axis, the modules can still join together by carrying out more complex movements involving having the matrices change the positions of these modules and other modules that are adjacent to them in such a way that they can join together.

# BASIC MODULE TRANSFERS, MOVEMENTS, AND CONNECTIONS

A module can be transferred around in a matrix of modules by pivoting and telescoping the legs, and connecting and disconnecting with the connecting plates. In Figure 4a, below, modules A-E are connected together to form a two dimensional matrix; in this matrix, module A is initially positioned in row 1, column 2, and its bottom leg connects to the top leg of module C, positioned in row 2, column 2. Module C is connected to and supported by the rest of the modules in the matrix through its bottom, left, and right legs. In the following figures, the legs for module A are drawn in different colors to illustrate their location in the movement as the module progresses from one position to another.

It is possible to transfer module A from row 1, column 2 to row 1, column 3, along the top surface of the rest of the matrix. Figure 4b, below, shows the same modules in Figure 4a, except that module A has been tilted by pivoting its bottom leg and the top leg of module C. The leg to the right of module A and the top leg of module D have also pivoted so the plates on the legs of these modules can connect to each other. The legs that pivot can also be extended so the connecting modules can reach each other, if necessary.

By reversing the motion of moving module A with module C and connecting it to module D, that is, disconnecting module A from module C and pivoting it upright with module D, module A is now in row 1, column 3; this is shown in Figure 4c, below.

It is also possible to pivot a module around a corner of the matrix, such as with module A around module D from its top to its right side (position row 1, column 3 to position row 2, column 4). Figure 4d, below, shows the legs that connect modules A and D (the top leg of module D and the bottom leg of module A from Figure 4c) pivoted to allow the right leg of module D to pivot and connect to the bottom leg of module A (the same leg that was on the right side of module A in Figure 4c.), which was also pivoted into position for connecting to the leg from module D.

Finally, just as reversing the movement pattern halfway through the process of transferring module A from position row 1, column 2 to position row 1, column 3 could be used to accomplish that process in Figures 4a, 4b, and 4c, the same concept can also be applied to transferring module A from position row 1, column 3 to position row 2, column 4. Figure 4e shows the final result with module A in position row 2, column 4.

At this point it’s trivial to determine that module A can travel all the way around the rest of the matrix of modules B, C, D, E, F, and G, back to its initial position, by repeatedly applying the basic techniques of transferring a module along a row from one column to another, as well as around corners. Transferring a module vertically from one row to another in the same column involves using the same technique as transferring a module from one column to another column in the same row. In general, any module can be positioned to any other location to reshape either a 2-dimensional or 3-dimensional matrix consisting of any number of modules into any combination of connected patterns. Figure 4b shows how the modules in a matrix are not limited to being connected in a rectangular cycle arrangment (e.g., the way modules B, C, E, and F form a cycle); they can also be connected in a triangular cycle arrangement (e.g., the way modules A, C, and D form a cycle). Notice, also, that in Figure 4d modules A and D are connected together with two pairs of legs each; by using this feature it is possible to connect a series of several modules in this way, to double the tension strength. In a 3-dimensional matrix of modules, up to three pairs of legs can be connected between modules to triple the tension strength in a series of modules that are connected together in this way, since there are 6 legs for each module (instead of just 4 legs for each module as in a 2-dimensional matrix).

# DETERMINING THE DISTANCE BETWEEN THE CENTERS OF TWO CELLS CONNECTED TOGETHER

The geometric structural topology of a full cell consists of it having the following fundamental characteristics:

1. the distance between the center of a cell and the point at which each leg pivots is a constant value
2. for each cube face, the distances between the center of a cell and the point at which each leg pivots are all equal to each other
3. for each cube face, the line segment between the center of a cell and the point at which each leg pivots is perpendicular to the line segments corresponding to each of the four adjacent cube faces

Many other characteristics about the topology of a cell can be derived from these fundamental characteristics and geometry. An essential key for a mesh of cells to form into objects is knowing its fundamental geometric structural topology and the distance between the centers of two cells connected together; this can be derived given that the following information is known about the modules in each cell that connect the two cells together:

1. the length of the telescoping leg (this value is a variable within a fixed range)
2. the angle a leg is pivoted, given as a value from a position perpendicular to the face of the cube; the absolute value of the angle a leg is pivoted will not be more than somewhere in the vicinity of 60 degrees
3. either the actual connecting plate relative angle of rotation or the virtual connecting plate relative angle of rotation (i.e., derived from the effect of pivoting a leg about both of its cube face-oriented pivoting axes)

Figure 20a shows two cells connected together using a similar illustration convention as shown in Figure 12a from the article Geometrical Cell Dimension Limits and Requirements.

Figure 20b shows the same two cells connected together from Figure 20a including some 2-dimensional cartesian axes, labels for certain points, and labels for distance between these points. For the cartesian axes, the y, y’, and y” (from here on a dimension shall be generalized in the form w(n)) axes are all parallel and aligned along the y-dimension origin; the x(n) (for a 3rd dimension, shown in Figure 20e) and z(n) axes are all parallel to their corresponding dimensions. The legs that connect two cells together are always parallel and aligned about their axes; in Figure 20b they are aligned parallel to the y(n) axes and with the z(n) axes origins. The z’-axis is aligned to be centered at point C. The z-axis and z”-axis are arranged to be placed in alignment with the leg pivoting points B and D, respectively. In this case, both legs have pivot angle values of 0. Points A and E are the centers of their corresponding cells. Points B and D are the points at which each corresponding leg pivots. Point C is the center of the two connecting plates that are connected together. The distance between points A and B is the constant value a, which is equal to the distance between points D and E; this value corresponds to the distance described by the first list shown above, the fundamental geometric structural topology characteristics. The length of the telescoping leg from the left cell is b, which is the distance between the points B and C. The length of the telescoping leg from the right cell is c, which is the distance between the points Cand D. These leg lengths correspond to the first piece of information from the second list shown above.

In this case, determining the distance r between the centers of each cell is straighforward:

(Formula #1)

In Figure 20c, the labelled points and corresponding distance labels of Figure 20b are shown without the cell body diagram illustrations and with one of the legs pivoted an angular value of α.

In this case the distance r between the centers of each cell is computed using the following equations and formula:

(Formula #1a)
(Formula #1b)
(Formula #2)

In Figure 20d, both legs are pivoted about parallel normal vectors, meaning all the labelled points (A, B, C, D, and E) lie on the same plane, thus only a 2-dimensional diagram is necessary for analysis; the second leg is pivoted with an angular value of β.

For this situation, calculating the distance r between the centers of each cell needs to be computed using the following equations and formula (Equation # 2b was plugged in) by either changing the definition of the angles to take a polar coordinate format about their corresponding origins, instead of being an absolute value format, or switching from plugging in Equation # 2b to plugging in Equation # 2c into Formula # 3 when the centers of the cells are on opposite sides of the z(n) axes from each other.

(Formula #2a)
(Formula #2b)
(Formula #2c)
(Formula #3)

In Figure 20e, not only are both legs pivoted about the y(n) axes, but also not all the labelled points (ABCD, and E) have to lie on the same plane, thus a 3-dimensional diagram is necessary for analysis. The difference between Figure 20d and Figure 20e can be interpreted as a result of at least one of two reason:

1. Connecting plate rotation took place between the two connecting plates connecting the cells together (at point C).
2. Pivoting of one of the legs connecting the two cells together about the x(n)-y(n) plane took place.

The actual connecting plate relative angle of rotation or the virtual connecting plate relative angle of rotation (i.e., produced as a result of pivoting the legs) is the angular value f.

The following equations and formula are used to calculate the distance r between the centers of each cell for the case of Figure 20e. In this format the x(n)-component of the point A (the center of the cell on the left) is held fixed equal to 0. If necessary, as an intermediate step the modules can use the equations to derive calculated values and then re-apply those values to the formula using the same equations for calculating the distance r; this occurs when the pivot of the leg for the cell on the left is actually the resultant of the leg pivoting on both axes, when none of the legs on the faces adjacent to the legs connecting the cells together lie on the same plane, or when there is both leg pivoting and plate rotation. This principle is true for the apparent discrete cases of Figure 20d where all the labelled points (ABCD, and E) happen to lie on the same plane; all that is necessary for this situation is to utilize the 3-dimensional approach for Figure 20e. It is not necessary to be concerned about whether or not the centers of the cells are on opposite sides of the z(n) axes from each other, as it was for the case of Figure 20d, because this is resolved by having equations for all 3 dimensions.

(Formula #3a)
(Formula #3b)
(Formula #3c)
(Formula #4)

In general, only Formula # 4 is needed to calculate r for all cases. By knowing the geometric structural topology of the cells, and for every cell the angle values α, β, f, the values for lengths a, b, c, and the distance between the centers of any two two cells r, a mesh of cells has all the information about cell positions and states that it needs for knowing the state of its structure, and it has the geometric information that is necessary for changing its shape.

# SENSOR-BASED DISTANCE DETERMINATION FOR CENTERS OF TWO CELLS CHAINED THROUGH A THIRD CELL

Certain lengths and angles can either be measured with sensors or are known constants that can be used to calculate the distance between the centers of two cells connected to a middle (third) cell such that they are chained in series. In order to determine the distance, each involved module within each cell would have to perform calculations on shared readings from both its own sensors and sensors on the other modules that are involved. In Figures 22a and 22b, 3 cells A, B, and C are drawn as three-dimensional green wire frames to represent the core of the cells (i.e., the cube-shaped region that doesn’t include the telescoping legs and connecting plates). The red numbered lines represent the measureable sections, they are either the distance from the center of the cell cube to the center of the face on the cube surface (all are equal and constant), or the length of the telescoping leg (each one is a variable value). There are two ways that 3 cells can be chained in series. One is shown in Figure 22a, where two cells (A & C) are connected to the middle cell (B) on opposite sides through it along the y-axis. In either case, the numbered red lines are:

1. Cell cube center to center of face for cell A connected to 2
2. Leg of cell A connected to 1 and 3
3. Leg of cell B connected to 4 and 2
4. Cell cube center to center of face for cell B connected to 3 and 5
5. Cell cube center to center of face for cell B connected to 4 and 6
6. Leg of cell B connected to 5 and 7
7. Leg of cell C connected to 6 and 8
8. Cell cube center to center of face for cell C connected to 7

The lengths of the “cell cube center to center of face” sections are all equal to each other in length L (i.e., L1 = L4 = L5 = L8 = L). For simplicity, all angle values written in formulas in this articleare limited to a single-cycle range of angle (0 ≤ angle < 2π).The other is shown in Figure 22b, where two cells (A & C) are connected to the middle cell (B) on adjacent sides through it along the positive y-axis, positive z-axis, and the x-y-z axes origin (where the 3 axes intersect).

In Figure 22c, the focus is on only some of the red lines, sections 1, 2, and 3 (or alternatively, sections 6, 7, and 8), in Figures 22a and 22b. Vector r1 (line segment OA) corresponds to sections 2 and 3 in Figures 22a and 22b, and vector r2 (line segment AC) corresponds to section 1 in Figures 22a and 22b. Points C1 and C2 are actually two possible graphical representations of a single point C, which is the center of the cell cube. The actual sensors are the universal joint horizontal and vertical angles, the connecting plate rotation angle, and the telescoping leg length. Readings from these sensors can be used as input values in conversion formulas as part of the process to perform the distance calculations. Since θ corresponds to the pivot angle of the leg, and the leg pivots on the face of the cell cube, it is never going to be more than a magnitude of 90° (i.e., 0 ≤ |θ| ≤ π/2).

The following equations written in terms of r are 3-dimensional vector and magnitude formulas:

(Formula #1)
(Formula #2)

Although r1 = L, r1 will remain in the equations for clarity. From Figure 22c, the following equations can be derived:

(Equation # 1)
(Equation # 2)
(Equation # 3)
(Equation # 4)
(Equation # 5)

If  1 = 0 and  2 = 0, the following equations can be derived and used.

(Equation # 6)
(Equation # 7)
(Equation # 8)
(Equation # 9)
(Equation # 10)

The following two equations are independent of any angle value 2.

(Equation # 11)
(Equation # 12)

The length of r0 can be calculated without θ1, using only r1, r2, and θ2, but ry and rz cannot.

(Equation # 13)

The following two equations apply only to point C1, not C2.

(Equation # 14)
(Equation # 15)

To make it easier to analyze the diagram in Figure 22c, some of the sections are redrawn in Figure 22d (Cartesian diagram on left side). The separate red-green-blue triangle in Figure 22d on the right side is a diagram of vector OB perpendicular to the plane of the diagram for arbitrary values of  2.

From Figure 22d, the following equations, which are independent of  2, can be derived:

(Equation # 16)

If  1 = 0, the following equation can be derived.

(Equation # 17)

For any angle value 2, in the y-z plane:

(Equation # 18)
(Equation # 19)
(Equation # 20)

In red-blue-green triangle figure:

(Equation # 21)
(Equation # 22)

For  1 = 0 and arbitrary values of  2:

(Equation # 23)

To generalize, the ρ-variables in the following equations are fixed to the formulas.

(Equation # 24)
(Equation # 25)
(Equation # 26)

For any value of 1, Figure 20e shows the necessary components for continuing the geometric analysis to derive the necessary formulas. The Cartesian diagrams with the green wire frames are based on Figure 22c for representing arbitrary values of 1 and 2. The blue triangle involves lengths ρz and ρx, and introduces angle β to cover the situation of 1 = 0 and arbitrary values of 2. For 1 = 0, angle β is basically a function of 2 to allow for calculating α as a function of angles 1 and β; all that is necessary is to add the two angles.Based on Figure 22e, the following equations can be derived:

(Equation # 27)
(Equation # 28)
(Equation # 29)
(Equation # 30)
(Equation # 31)
(Equation # 32)
(Equation # 33)

Now from the previous equations, the following formulas can be derived:

(Equation # 34)
(Equation # 35)
(Equation # 36)
(Equation # 37)

To calculate the distance d from the origin O to the center of cell A (shown in Figure 22a), all that is necessary is to include section 4 in the equations:

(Equation # 38)
(Equation # 39)
(Equation # 40)

The previous three equations are re-written to include both sections 1-4 and 5-8 in Figure 22a. The next three are for sections 1-4:

(Equation # 41)
(Equation # 42)
(Equation # 43)

The next three equations are for sections 5-8:

(Equation # 44)
(Equation # 45)
(Equation # 46)

The following equations result from adding the last six equations together, to get the formulas for the distance D from the center of cell A to the center of cell C in Figure 22a:

(Equation # 47)
(Equation # 48)
(Equation # 49)

The following three equations represent sections 5-8 in Figure 22b:

(Equation # 50)
(Equation # 51)
(Equation # 52)

# CELL STRUCTURE FORMING PATTERN POSSIBILITIES

There are several ways a module can be attached to other modules to form a partial cell, full cell, or inverse cell. In Figure 23a, the different side views of a single module is shown in various colors. The leg and connecting plate assembly is a darker blue, and the rest of the module is a different color for the type of side. For the opposite sides that are symmetric, the longer (lengthwise) side are green, and the shorter (widthwise) sides are red. The surface on which the leg mounts is light blue, and the opposite side is yellow. In this diagram, the truncated pyramid-shaped surface shown in this article: Module Connection Arrangement Patterns of a Cell is omitted, but the yellow side corresponds to the side on which it is mounted.

One type of connection of a module is to another single module via the connecting plates. This is shown in Figure 23b and is referred to as an “inverse cell.” The idea is that in the same way a mesh or matrix can be made of cells, and cells are made using modules, a mesh or matrix can also be made of inverse cells, and inverse cells can also be made using modules. In this figure, different colors do not necessarily signify different modules. Both the legs and the connecting plates of both modules are dark blue. On the left side of the line indicated by a vertical series of gray dashes is one module with a green lengthwise side shown, and on the right the other module with a red widthwise side shown. It is not necessary for the opposite types of side indicated by red or green to be aligned with each other from the two modules or to be aligned in general. It is only shown this way to make it easier to identify each module. The different resulting patterns will be categorized with abstract codes as connection types. A single module will be categorized simply as M. The resulting module structure in Figure 23b will be categorized as connection type T.

An expression describing the connection will also be given in the forms F → R or R ← F, meaning “F can form R,” where R is the resulting form and F is an abstract representation of “X + Y,” which means the joining of object X to object Y, and it is commutative (i.e., X + Y and Y + X are interchangeable). The objects X and Y each can be either a module M or a partially constructed cell R.

Expression: M + M → T

Another type of connection of a module M to another single module M is via the rectangular bases, which is to connect one along one of the green surfaces side (not shown) to the yellow widthwise side (not shown) of another in the type of orientation and alignment shown in Figure 23c. The smaller square to the right is an abstract icon representing the view angle of the figure with corresponding color pattern (shown for future reference). In this figure as well, different colors do not necessarily signify different modules. Both the legs and the connecting plates of both modules are dark blue. In this case it is necessary for the opposite types of side indicated by red or green to be aligned with each other from the two modules to be aligned. The different resulting patterns will be categorized with abstract codes as connection types. A single module will be categorized simply as M. The resulting module structure in Figure 23c will be categorized as connection type A. A formula describing the connection will also be given, for instance:

Expression: M + M → A

In the remaining part of this article the side views of the dark blue leg and connecting plate will be omitted from the diagrams and will only be limited to connections via the rectangular bases.

A single module can also be connected to a group made up of 2 to 5 modules connected together via their rectangular bases. Figure 23d shows one way of connecting a module M to each of a pair of modules A resulting in a partial cell in which all three modules share a common corner. The icon on the left is shown as a different angle than the figure image, and sometimes the view of the module in the back is obstructed by the one in front, so the gray circle with the smaller icons represent the side of the viewing direction of the figure (not present if viewing directions are the same) to reveal the presence of modules in the back on the icon. This resulting module structure will be categorized as connection type Bx.

Expression: A + M → Bx

Figure 23e shows a reflective way a of connecting a module M to each of a pair of modules A resulting in a partial cell in which all three modules share a common corner, which is a mirror image of the version in Figure 23d. This resulting module structure will be categorized as connection type By.

Expression: A + M → By

Instead of adding a single module M to a pair of modules resulting in a common corner, it can be added to one of the modules to be on the opposite side of the other module of an A structure. Figure 23f shows one version of this type of structure, categorized as connection type Ux.

Expression: A + M → Ux

Figure 23g shows another version of the type of structure similar in arrangement pattern to the structure of Figure 23f using an M and A structure, categorized as connection type Uy. These two from Figures 23f and 23g are opposites to each other in that they can join together to form a full cell.

Expression: A + M → Uy

Another way in which a single module M can be connected to a group of multiple modules connected together via their rectangular bases is shown in Figure 23h, which is to connect it to a set of 3 modules in a Ux structure. This is one version that results in this pattern where the 4 modules joined together results in a 2D alternating cycled pattern, to be categorized as connection type O.

Expression: Ux + M → O

The other way to join a single module M to 3 other modules to result in connection type O is shown in Figure 23i, by using a Uystructure.

Expression: Uy + M → O

Another way to connect a module M to a set of 3 modules in a Ux structure is shown in Figure 23j. This is one version that results in a pattern where the 4 modules joined together do not result in the 2D alternating cycled patterns of Figures 23h and 23i, to be categorized as connection type G.

Expression: Ux + M → G

Figure 23k also shows another way to connect a module M to a set of 3 modules, which are in a Uy structure, resulting in a G structure.

Expression: Uy + M → G

Another way in which a single module M can be connected to a group of multiple modules connected together via their rectangular bases is shown in Figure 23m, which is to connect it to a set of 4 modules in a G structure. This one is to be categorized as connection type Q.

Expression: G + M → Q

Figure 23n also shows another way to connect a module M to a set of 4 modules in a G structure which also results in a connection type Q.

Expression: G + M → Q

Figure 23o shows a way to connect a module M to a set of 4 modules in an O structure pattern for form a Q structure.

Expression: O + M → Q

Figure 23p shows a full cell, categorized as connection type C, made by connecting a module M to a Q structure.

Expression: Q + M → C

Forming a structure is not limited to being possible by adding just a module M to another structure. Two structures A and A can be joined together to form a structure. A structure O made by joining two structures A and A is shown in Figure 23q.

Expression: A + A → O

A structure G can be made by joining two structures A and A as shown in different perspectives in Figures 23r and 23s. Since it might not be simple to determine that different views are the same structure forming connection pattern, the modules are labeled A, B, C, and D to indicate which ones in Figure 23r can correspond to the ones in Figures 23s.

Expression: A + A → G

A structure Q can be made by joining two structures A and By as shown in Figure 23t.

Expression: A + By → Q

A structure Q can also be made by joining two structures A and Bx as shown in Figure 23u.

Expression: A + Bx → Q

A full cell structure Q can be made by joining two structures A and Ux as shown in Figure 23v.

Expression: A + Ux → Q

A full cell structure Q can also be made by joining two structures A and Uy as shown in Figure 23w.

Expression: A + Uy → Q

A full cell structure C can be made by joining two structures A and G as shown in Figure 23x.

Expression: A + G → C

A full cell structure C can also be made by joining two symmetrically opposite B structures Bx and By as shown in Figure 23y.

Expression: Bx + By → C

A full cell structure C can also be made by joining two symmetrically inverse U structures Ux and Uy as shown in Figure 23z.

Expression: Ux + Uy → C

The following table summarizes the expressions, and groups each connection F with matching results R or vice versa by separating each possible type of joining connection or result by a “|” symbol (meaning “or”), listed by number of modules in the resulting cell. If the subscript is omitted it means all versions are included.

Number of Modules Expression
2 M  + M   → A  | T
3 A  + M   → B  | U
4 O  | G   ← U + M  | A + A
5  Q   ← G + M  | O + M  | A + B  | A + U
6  C   ← Q + M  | A + G  | Bx + By  | Ux + Uy

The following table summarizes the distinct possible “|” separated R result patterns arranged in order of number of modules in each joining part.

Structure Connections Resulting Pattern Possibilities Notes
1: + 1 = 2 A  | T only one possible way to form each
+ 2 = 3 Bx  | By  | Ux  | Uy only one possible way to form each
+ 3 = 4 O  | G two symmetrically complementary possible ways to form each
+ 4 = 5 Q three possible ways to form, two are complementary to each other
+ 5 = 6 C only one possible way to form
2: + 2 = 4 O  | G only one possible way to form each
+ 3 = 5 Q four possible ways to form, 2 pairs of symmetrical compements
+ 4 = 6 C only one possible way to form
3: + 3 = 6 C two symmetrically complementary possible ways to form