• Internal Code :
  • Subject Code : NMBR140
  • University : Australian Catholic University
  • Subject Name : Mathematics

Transcript

Hi. I am happy to introduce myself as Student name. As a part of my course name, I have created a video presentation that explains the importance of mathematical modelling. 

Most of the objects encountered in our daily lives have either a regular shape or an irregular shape.  This fits for both natural or human-made objects. For example, bee-hive is made up of regular hexagons while clouds have a very irregular shape. Hence it is very important to understand geometry and mathematical modelling helps in this way. 

A well-described approach to overcome this challenge is to make the students actively engaged in topics covering geometric and spatial thinking and by helping them understand the relevance in real-life. Introducing geometric shape, form and structure, geometric measurement etc. by creating models and visualizations related to geometry help the young children to perceive the world around in a better way. Thus, the mathematical modelling of various geometric visual tools helps in understanding geometry efficiently.

In this slide, a polyhedron called Truncated cube is presented. A shape that is formed by a collection of shapes is called a polyhedron. The polyhedron presented below is a uniform polyhedron, called a “Truncated Cube”. This is a collection of two regular shapes. The two shapes joined to form a truncated cube are:

  1. Octagon

  2. Triangle.

The truncated cube has 14 faces and 24 vertices altogether.  There are 6 regular octagonal faces and 8 triangular faces. The triangular faces are made by equilateral triangles. The polyhedron is called a truncated cube because a cube is truncated by cutting off the tips to form a new shape.

 

 

The dihedral angles in the polyhedron can be either an oct-oct angle or an oct-tri angle. The oct-oct angles are right angles at 90 degrees and the oct-tri angles are at 125 degrees and 16 minutes. 

It is an Archimedean Solid shape. Archimedean solids are all solid shapes that contain two are more regular shapes that have identical vertices. These regular polygons are identically arranged around themselves to give rise to a regular polyhedron. 

 

In this slide, a regular convex polyhedron called Rhombicuboctahedron is shown. It has 8 triangular faces and 18 square faces. Every square is connected with 3 equilateral triangles around each of its 24 vertices. These regular polygons can be divided into 

  1. a top and bottom square cupola shapes made up of alternating squares and equilateral triangles and 

  2. a middle belt that is made up of all squares. 

 

This shape is an Archimedean solid. It can be rotated along any two vertices and still the same shape can be obtained, making it a perfect Archimedean shape. It means that the sum of interior angles does not change by rotation along the same plane and hence the object has perfect symmetry.

The next slide shows the net wire diagram that can be folded into a Rhombicuboctahedron. Subsequently, it can be seen that the same image is obtained even after moving from one vertex to another. Thus, it could be understood that it has perfect symmetry.

 

On the other hand, the second shape shown in this slide is a pseudo Rhombicuboctahedron and is not Archimedean. This is because of the slight change made in the way the triangles and squares are attached. In the bottom and the top, the square cupolas of the rhombicuboctahedron are rotated slightly and attached so that the side of a triangle meets the side of a square. This rotation gave rise to a new polyhedron called the pseudorhombicuboctahedron or Miller’s shape. It is also called an elongated Square Gyrobicupola. This is not an Archimedean solid because when rotated, it does not lead to the same shape always.

The position where the squares of the middle belt meet the equilateral triangles of the square cupolas is altered. This polyhedron, though resembles the rhombicuboctahedron superficially, is a different polyhedron. This alternate shape is not Archimedean as it has lost perfect symmetry property.

So, though both the shapes are made using the same two regular polygons, the way the net is arranged or the way the polyhedrons are formed differ. 

References

Boag, T. (1979). On Archimedean Solids. Mathematics Teacher.

Burt, J. L., Elechiguerra, J. L., Reyes-Gasga, J., Montejano-Carrizales, J. M., & Jose-Yacaman, M. (2005). Beyond Archimedean solids: Star polyhedral gold nanocrystals. Journal of Crystal Growth. https://doi.org/10.1016/j.jcrysgro.2005.09.060

Chen, E. R., Klotsa, D., Engel, M., Damasceno, P. F., & Glotzer, S. C. (2014). Complexity in surfaces of densest packings for families of polyhedra. Physical Review X. https://doi.org/10.1103/PhysRevX.4.011024

Ouyang, P., Zhao, W., & Huang, X. (2015). Beautiful math, part 5: Colorful archimedean tilings from dynamical systems. IEEE Computer Graphics and Applications. https://doi.org/10.1109/MCG.2015.135

Schein, S., & Gayed, J. M. (2014). Fourth class of convex equilateral polyhedron with polyhedral symmetry related to fullerenes and viruses. Proceedings of the National Academy of Sciences of the United States of America. https://doi.org/10.1073/pnas.1310939111

Wohlhart, K. (2014). Twisting towers derived from Archimedean polyhedrons. Mechanism and Machine Theory. https://doi.org/10.1016/j.mechmachtheory.2014.05.001

Zhang, L., & Schmitt, W. (2011). From platonic templates to archimedean solids: Successive construction of nanoscopic {V16As8}, {V16As10}, {V20As8}, and {V24As8} polyoxovanadate cages. Journal of the American Chemical Society. https://doi.org/10.1021/ja2024004

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