By adapting a previously written percolation model in C, the threshold probabilities for square, triangular, and cubic lattice types were confirmed. An algorithm to count the distribution of cluster sizes at a variety of percolation probabilities was developed, and the expected trends towards the so called infinite cluster was achieved. An equivalent bond percolation model was adapted to the original site algorithm, and by treating occupied bonds as springs, a total compression trend for the model was constructed, which implied that structures under the boundary conditions that were imposed does not have behavior that changes the total compression constant significantly at the percolation threshold.\\
This paper implements Simultaneous Localization and Mapping (SLAM) technique to construct a map of a given environment. A Real Time Appearance Based Mapping (RTAB-Map) approach was taken for accomplishing this task. Initially, a 2d occupancy grid and 3d octomap was created from a provided simulated environment. Next, a personal simulated environment was created for mapping as well. In this appearance based method, a process called Loop Closure is used to determine whether a robot has seen a location before or not. In this paper, it is seen that RTAB-Map is optimized for large scale and long term SLAM by using multiple strategies to allow for loop closure to be done in real time and the results depict that it can be an excellent solution for SLAM to develop robots that can map an environment in both 2d and 3d.