Popularization of Mathematics - the Queen of the Sciences by Demonstrating its Daily Implementation and Visualization

We will review a number of areas of mathematics and examples of their applications.
a. Differential-Integral Calculus: Fibonacci Series - An ascending series of numbers in which the quotient of the two differences in the last neighbor numbers converges to a golden ratio, which is used in architecture. Mathematically, the concept of a limit is illustrated here.
b. Logic: Syllepsis is presented here, which is defined as a mistake in drawing conclusions that lead to a contradiction - a suitable arithmetic example is given. This is an amusing illustration to learn logic.
c. Numbers Theory: Prime Numbers - An illustration of a spiral of Ulam showing the location of the Prime Numbers on the x-axis [2].
d. Algebra: The concept of a Group might be illustrated with the help of the Hungarian Cube.
e. Graph Theory: The Euler Path, Introducing the “Königsberg Bridges” Problem, Solving Navigation Problems - the “Traveling Salesman” Problem.
f. Numerical Analysis: A visual illustration of the iterations needed to find the zero of a function using the Tangent Method.
g. Theory of Variations: Search for an optimal function that describes the shape of a hanging rope; its shape is formed under the constraints of gravity. These shapes are represented in decorative chains that are popular in architectural design.
h. Combinatorics: Concepts such as permutations and combinations are illustrated by the Sierpiński Triangle.
i. Approximation Theory: It represents, with the help of the Stirling formula, an approximated function of the factorial function.
j. Probability: It might be represented by the concept of randomness in conjunction with the visualization of a casino operation, for example [3].
k. Set Theory: A Universal Set leading to internal contradictions – demonstrated by the “Barber Paradox” and the concept of the countable sets may represent this topic.
l. Topology: Different surfaces and spaces - Demonstration of the “Klein Bottle”, in which there is a transition from the bottle envelope to its interior (Figure 1), emphasizing the importance of the visuality.

Figure 1:Transparent KLEIN bottle illustrating its entrance, showing the path from the envelope to its interior.

1. Ophir D (2011) DDDL: A descriptive, didactic and dynamic programming language. Israeli-Polish Mathematical Meeting, Łódź, Poland.
2. Ophir D (2017) A sociable robot as a language teacher. The Israeli Association for Literacy and Language, The Open University, Raanana, Israel.
3. Gardner M (1971) Mathematical games from scientific American. Freeman and Company, San Francisco, USA.
4. Steinhaus H (1989) The mathematical kaleidoscope. Polish Pedagogical Edition, Warsaw, Poland.