2 bonded carbon atoms

Notice that each covalent bond could rotate one hundred and twenty degrees to the next favorable spot.  Each carbon thus has potentially three favorable states. 

Consider two adjacent carbon atoms.  The hydrogen atoms will be in an energetically favorable position when they are in the staggered so called "trans or anti" configuration. 

 The staggered conformation is also known as the anti-conformation, where the hydrogen atoms of one carbon are positioned away from the hydrogen atoms of the adjacent carbon, resulting in minimum steric hindrance. In contrast, in the eclipsed conformation, the hydrogen atoms of one carbon are positioned directly behind the hydrogen atoms of the adjacent carbon, leading to maximum steric hindrance. The staggered (anti-conformation) is the most stable conformation for most organic molecules, including alkanes, because it minimizes the repulsion between electron clouds of adjacent atoms. 

 We now add a few more lines to complete the cube. 

 Consider a set of two bonded carbon atoms that can each exist in one of three states. We therefore have 2**3 or eight possible states which can be labeled as 000, 001, 010, 011, 100, 101, 110, 111. To go from 000 to 001 requires the rotation of a single bond. To go from 000 to 110 requires 2 rotations etc.  A cube balanced on a single corner where the eight corners are the states and the twelve edges are the transitions between states, can depict this situation.   

The cube analogy is a visual representation of the state transitions between the two bonded carbon atoms with three possible states each. Each corner of the cube represents one of the eight possible states, and the edges represent the transitions or rotations between these states.

In this case, the cube can be viewed as a graphical representation of a state transition diagram or a state transition table, commonly used in computer science and control systems. The transitions between states can be depicted as directed edges connecting the corresponding corners of the cube.

For example, starting from state 000, a transition to state 001 requires a rotation of a single bond. This transition can be represented by an edge connecting the corresponding corners labeled 000 and 001. Similarly, transitions from state 000 to 110, which requires two rotations, can be represented by an edge connecting the corresponding corners.

By visualizing the state transitions in this way, you can easily understand and analyze the possible transformations between the different states of the carbon atoms. It provides a convenient representation to study the dynamics and behavior of the system, particularly in terms of how changes in bond rotations can lead to different states.

This cube analogy can serve as a useful tool for understanding and designing molecular systems, especially when considering the interplay between multiple states and transitions. It allows for a visual exploration of the relationships between different states and provides insights into the potential dynamics and behaviors of the system.

Labeling the edges and their inverses :

Edge Labels: E1: 000 -> 001 E2: 001 -> 011 E3: 011 -> 010 E4: 010 -> 000 E5: 000 -> 100 E6: 001 -> 101 E7: 011 -> 111 E8: 010 -> 110 E9: 100 -> 101 E10: 101 -> 111 E11: 111 -> 110 E12: 110 -> 100

Inverse Edge Labels: E1': 001 -> 000 E2': 011 -> 001 E3': 010 -> 011 E4': 000 -> 010 E5': 100 -> 000 E6': 101 -> 001 E7': 111 -> 011 E8': 110 -> 010 E9': 101 -> 100 E10': 111 -> 101 E11': 110 -> 111 E12': 100 -> 110

These labels represent the transitions between the different states of the two bonded carbon atoms. Each edge represents a specific rotation or transition from one state to another, and its label indicates the initial and final states of that transition.

The inverses of the edges represent the reverse transitions, allowing you to go back from the final state to the initial state. They provide a way to undo the rotation or transition and revert to the original configuration.

These labeled edges and their inverses provide a systematic way to represent the possible state transitions and reversals in the system of the two carbon atoms. They form the basis for analyzing and understanding the dynamics of the molecular system and the relationships between different states.

The set of all sequences of moves on the cube can form a group. Let's explore the properties:

1. Closure: The property of closure holds because when you combine two sequences of moves (edges), the result is another sequence of moves (edge). This means that performing one sequence of moves followed by another sequence of moves will always yield a valid sequence of moves on the cube.
2. Associativity: The property of associativity holds because the order in which you combine sequences of moves does not affect the final result. Whether you group the moves as (E1 * (E2 * E3)) or ((E1 * E2) * E3), you will end up with the same resulting sequence of moves on the cube.
3. Identity Element: The "do nothing" sequence serves as the identity element in the group. This means that if you perform the "do nothing" sequence followed by any other sequence of moves, the result will be the same as the original sequence. The "do nothing" sequence does not change the state of the cube.
4. Inverses: For every sequence of moves (edge), there exists an inverse sequence of moves (inverse edge) that can undo the effects of the original sequence. This means that if you perform a sequence of moves and then perform its corresponding inverse sequence of moves, you will return to the original state of the cube.

Based on these properties, it can be concluded that the set of all sequences of moves on the cube forms a group. This group exhibits closure, associativity, the existence of an identity element, and the existence of inverses for every element.

Coincidentally we could relabel the above cube as follows:     

To represent the relation “containment” of one subset in another for the partially ordered set of all subsets of the set (a,b,c).  

Represent the relation of divisibility for the partially ordered set (1,2,3,5,6,10,15,30) as follows:  

If we consider the partially ordered set (1, 2, 3, 5, 6, 10, 15, 30) and want to represent the relation of divisibility on a cube, we can assign each number to a vertex of the cube. The cube will have eight vertices, and each vertex represents a number from the set.

To indicate the divisibility relation, we can draw edges between vertices where one number divides another. For example, we can draw an edge between vertices 1 and 2 since 1 divides 2, and an edge between vertices 1 and 3 since 1 divides 3. Similarly, we can draw edges for other pairs of numbers that have a divisibility relationship.

The resulting cube with edges connecting the vertices representing the numbers would visually represent the divisibility relation within the set (1, 2, 3, 5, 6, 10, 15, 30) as a partially ordered set.

The following color coded cube demonstrates two more potential useful properties: (i) three sets of four orthogonal edges and (ii) a bipartite set of corners.  

Here are a few more mathematical structures and concepts that can be represented on the cube:

1. Binary Operations: Each edge of the cube can represent a binary operation between the numbers at its endpoints. For example, addition, subtraction, multiplication, or any other binary operation can be associated with the edges of the cube. The result of applying the binary operation to the numbers at the endpoints of an edge can be represented by the corner where the edge terminates.
2. Equivalence Relations: We can define equivalence relations on the cube by grouping corners that satisfy certain conditions. For example, we can group corners that represent numbers with the same remainder when divided by a specific modulus. Each group of equivalent corners would then represent an equivalence class.
3. Graph Theory: The cube can be used to represent various graph structures and properties. For example, the corners can represent vertices, and the edges can represent edges in a graph. The connectivity between vertices can be represented by the presence or absence of edges between the corresponding corners.
4. Geometric Transformations: Each corner of the cube can represent a point in three-dimensional space, and the edges can represent transformations such as translations, rotations, or reflections. By applying these transformations to the corners, we can visualize various geometric concepts and explore geometric properties.
5. Logic Gates: The corners and edges of the cube can be associated with logic gates such as AND, OR, NOT, XOR, etc. Each corner can represent the output of a logic gate based on the inputs represented by the edges connected to that corner.

These are just a few examples of how the cube can be used to represent different mathematical structures and concepts. The cube's vertices, edges, and faces provide a versatile framework for visualizing and understanding various mathematical ideas.

Intramolecular computation refers to computational processes that take place within a single molecule or molecular system. It involves utilizing the internal properties and interactions of the molecule to perform computation or information processing tasks.

Intramolecular computation can be achieved through various mechanisms and molecular components. Here are a few examples:

1. Conformational Changes: Molecules can exist in different conformations or structural states, and these states can be utilized to encode and process information. By manipulating the conformational changes of a molecule, such as through external stimuli or chemical reactions, computational operations can be performed. For example, a molecule with multiple stable conformations can represent different binary states, and transitions between these states can be used for logical operations.
2. Switchable Molecules: Switchable molecules possess the ability to toggle between different states or properties in response to external signals. This switching behavior can be harnessed for computation. For instance, photochromic molecules change their structure or properties upon exposure to light, enabling them to function as binary switches in computational processes.
3. Molecular Logic Gates: Logic gates are fundamental building blocks of digital circuits. In intramolecular computation, molecular logic gates can be designed and implemented using specific molecular systems. These gates utilize the molecular properties and interactions to perform logical operations, such as AND, OR, and NOT. By combining multiple molecular logic gates, more complex computations can be achieved.
4. Molecular Motors: Molecular motors are molecular-scale devices that can convert chemical energy into mechanical motion. They can be employed in intramolecular computation to perform specific tasks or manipulate information. For example, molecular motors can be used to move molecules or components within a system, enabling transport, sorting, or assembly processes.

Intramolecular computation is an emerging field with potential applications in areas such as nanotechnology, molecular robotics, and synthetic biology. It offers the advantage of working at the molecular scale and harnessing the inherent properties of molecules for computation. However, challenges remain in terms of designing reliable and controllable molecular systems, as well as integrating them into larger-scale computational architectures. Ongoing research aims to explore the capabilities and limitations of intramolecular computation and develop novel approaches for molecular-level information processing.