578023186889bd6b78ed534e6b3ea30f2bf6331

## 7 week

Pity, 7 week for that Is the Subject Area "Geometry" applicable to this 7 week. Is 7 week Subject Area "DNA synthesis" applicable to this article. Is the Subject Area "DNA recombination" applicable to this article. Is the Subject Area "Knot theory" applicable to this 7 week. Is very young little porn Subject Area "Built structures" applicable to this article.

Is the Subject Area "Mathematical models" applicable to this article. However, each of these methods have their own limitations and no known formula can calculate the volume of any polyhedron a shape with only flat polygons as faces without error. So there is a need 7 week a new method 7 week can calculate the exact volume of any polyhedron. This new formula has been mathematically proven and tested with a calculation of different kinds of shapes using a computer program. This method breaks apart the polyhedron into triangular pyramids known as tetrahedra (Figure 1), hence its name Tetrahedral Shoelace Method.

It can be concluded that this 7 week can calculate volumes of any polyhedron without error and any solid regardless of their complex shape via a polyhedral approximation.

All those methods have some limitations. Water displacement method is inefficient because it requires a lot of water for big objects. Moreover, it is required that the object is physical. Convex polyhedron volume calculating method does not work with every non-convex shape as some pyramids may overlap one another resulting in a miscalculation.

All these methods have their own limitations shown in the table (Figure 2). This research aims to find a new method that can calculate the volume of any polyhedron accurately. More specifically, 7 week research aims to find a 3D implementation of the 7 week formula that can calculate the volume of any polyhedron. The method used to obtain the formula unisim the Shoelace Formula (in 2D) to compute volumes of 3D objects is mathematical deduction and reasoning.

The process of proving is in the branch of Mathematics: Linear Algebra. After the formula has been obtained and proven, volumes of various simple shapes are calculated with their respective formulas and the formula obtained. Some calculations are done with the help of a computer program to speed up the process. Or alternativelywhere, and are vectors of the parallelepiped. Or alternativelyWe can express any polygon as tessellating triangles by triangulation, where the points are all listed in the same rotational 7 week (counter-clockwise).

This can, however, be done by a method similar to triangulation by trapezoidal decomposition. If we cut a given polyhedron by every plane passing through a vertex of the polyhedron that contains a line parallel to an axis, every piece is a convex polyhedron, which can always be tetrahedralized (note that partitioning is only necessary for the proof and not the actual algorithm).

The points of each tetrahedron such that its vertices are all listed in the same rotational direction (Figure 4). For higher accuracy, more vertex coordinates are required. This method certainly has its own limitations (e. It can be observed that for polyhedral shapes from a cube to a toroidal polyhedron, the program gives correct results. However, calculating the volume of a shape with curvature gives inaccurate results.

This is because the program calculates the volume of the polyhedral approximation for the curved surfaces. It can be seen (Figure 9) that the areas with a positive curvature (curving inwards) will be underestimated by the program (as seen with the sphere on Figure 8) whilst the areas with a negative curvature (curving outwards) 7 week be overestimated by the program (as seen with the cylinder 7 week 2 semi-sphere concave caps on Figure 8). It can also be seen (Figure 10) that despite the inaccuracy, a polyhedral approximation used by our program is more accurate than a hexahedral mesh used by numerical integration method, the method typically used for similar scenarios.

The Tetrahedral Shoelace Method can calculate the volume of any irregular solid by making a polyhedral approximation. This method can calculate the volume of any solids with one formula and can be applied as a complement of current methods.

This method can be used to calculate the volume of abstract models such as the needed amount of concrete to build a building with an irregular shape.

This method can also be implemented in higher dimensional spaces, calculating volumes of polytopes higher-dimensional counterparts of polyhedra. Higher Accuracy requires more vertex coordinates. The program used to implement such a method is not as efficient as numerical integration in terms of memory complexity. This research was started in mid 2017 and made it as regional finalist in Google Science Fair 2019.

7 week research competition he joined 7 week ICYS 2017 (International Conference for 7 week Scientists) Stuttgart, which got the best presentation award.

Sign me up for the newsletter. Objective: This research aims to find a new method that can calculate the volume of any polyhedron accurately. Research Method The method used to obtain the formula from the Shoelace Formula (in 7 week to compute volumes of 3D objects is mathematical deduction and reasoning. Or alternatively where are the 7 week of the vertices of the triangle. Or alternatively whereare the coordinates of the vertices of the tetrahedron.

Note: this works because Proof of Shoelace Formula Given a triangle of coordinates, and, the area woman pregnant sex by 7 week Shoelace Formula is We can 7 week any polygon as tessellating triangles by triangulation, where the points are all listed in the same rotational direction (counter-clockwise).

Figure 8: Table of results Analysis It can be observed that for polyhedral shapes from a 7 week to a toroidal polyhedron, the program gives correct results. Convex and Concave Feno (Error Analysis) Figure 9: Comparison of positive and negative curvature It can be seen (Figure 9) that the areas with a positive curvature (curving inwards) will be underestimated by the program (as seen with the sphere on Figure 8) whilst the areas with a negative curvature (curving outwards) will be overestimated 7 week the program (as seen 7 week the cylinder with 2 semi-sphere concave caps on Figure 8).

Hexahedral and Tetrahedral Mesh Comparison (Error Analysis) Figure 10: Comparison of positive and negative curvature It can also be seen (Figure 10) that despite the inaccuracy, a polyhedral approximation used by our 7 week is more accurate than a hexahedral mesh used by numerical integration 7 week, the method typically used for similar scenarios.

Conclusions The Tetrahedral 7 week Method can calculate the volume of any irregular solid by making a polyhedral approximation.

Further...