"Polytopes" Example Sentences
1. Polytopes are fascinating mathematical objects.
2. I am currently studying the properties of polytopes.
3. There are many types of polytopes, including cubes and pyramids.
4. The edges of polytopes can be straight or curved.
5. One of the most famous polytopes is the dodecahedron.
6. Mathematicians have classified polytopes based on their number of faces.
7. Non-convex polytopes are more complicated than convex polytopes.
8. It is possible to construct polytopes in higher dimensions, beyond three dimensions.
9. The Platonic solids are a group of regular polytopes.
10. The symmetry of polytopes is an important area of study in geometry.
11. The dual of a polytope is another polytope with certain properties.
12. Skew polytopes are polytopes that are not contained entirely within a single hyperplane.
13. Some polytopes can be used to model complex patterns in nature.
14. The study of polytopes has applications in fields such as computer science and physics.
15. The area of polytopes is one topic covered in geometry courses.
16. In higher dimensions, it becomes more difficult to visualize polytopes.
17. Convex polytopes have nice properties that make them useful in optimization problems.
18. The notion of a polytope can be generalized to include objects with curved surfaces.
19. The vertices of a polytope are the points where its edges meet.
20. One way to construct a polytope is to start with a lower-dimensional polytope and add cells to it.
21. The study of the rigidity of polytopes is an important area of research.
22. The properties of a polytope can be deduced from its combinatorial structure.
23. The concept of symmetry is closely related to the study of polytopes.
24. The volume of polytopes can be computed using advanced mathematical techniques.
25. The theory of polytopes has connections to group theory, topology, and other mathematical areas.
26. The study of polytopes involves understanding the relationships between their various parts.
27. Different types of polytopes have different numbers of faces and vertices.
28. The Cayley-Menger determinant is a formula used to compute volumes of polytopes.
29. Some polytopes, such as the stellated dodecahedron, have intricate patterns that make them beautiful to look at.
30. The study of polytopes is a rich and rewarding area of mathematics.
Common Phases
1.
Polytopes are geometric objects that have flat sides and corners.
2. A regular polytope is a polytope that has all sides and angles the same size.
3. The simplest
polytopes are the regular polygons and regular polyhedra.
4. A convex polytope is a polytope where a straight line between any two points on the surface lies completely within the polytope.
5. The five Platonic solids are the only convex regular polyhedra.
6. A non-convex polytope is a polytope where a straight line between any two points can lie outside the polytope.
7. Examples of non-convex
polytopes include star polyhedra and regular-faced Johnson solids.
8. The study of
polytopes is known as polytopology.
9. The higher-dimensional analogues of polygons and polyhedra are called
polytopes.
10. In higher dimensions, there are many different types of
polytopes, including simplices, hypercubes, and hypertetrahedra.