Pythagoras theorem states that, in a right triangle, the square of the hypotenuse is equal to the sum of the square of the other two sides.

The Pythagorean Theorem is the first nontrivial fact we encounter in geometry. The circular disk illustrates the theorem but does not prove it, specifically for the dimensions chosen in this model. So why would it apply to other dimensions? On the other hand, a puzzle is the basis for a mathematical proof. And we can show by logic that it will always apply.

The circumference-to-diameter ratio of a circle is a constant called the approximate ratio (π) with a value of 3.141592.

Since the width of the rectangle is equal to the radius of a circle, the rectangle’s length is equal to half its circumference and their areas are equal, this demonstrates the theorem that the area of a circle is:

½ x (radius) x (circumference).

The Circle: The set of points in a plane that are equidistant from a given point (center).

Ellipse: The set of points whose distance from two given points (foci) sums to a constant amount.

A rubber string takes the shortest path on the surface of the Earth between its two ends. This is the path (geodesic) that airplanes take. If you were to plot this path on a regular map, it would appear curved toward the poles, because planar maps distort distances.

In “two-dimensional” models, the lower paths meet at 120° angles. Three-dimensional problems require a deeper dive into math & There are many applications of this topic, from designing networks to understanding the membranes of living organisms.

Each point on the rod draws a horizontal circle in space. The circles drawn by the middle part of the rod are smaller than those drawn by the outer parts of the rod. Together, these circles form a narrow-waisted vertical “cylinder”.

The Soma cube is a solid dissection puzzle invented by Danish polymath Piet Hein in 1933 during a lecture on quantum mechanics conducted by Werner Heisenberg.

A) Robotics is an important and growing technology, and reaching elasticity is as much a mathematical challenge as it is a technical one. It makes us marvel at how well our bodies are meeting this challenge!;

B) Describing the state of a robot relies on the concepts of configuration space and degrees of freedom.;

C) Configuration space is a mathematical tool created to enable us to mathematically describe all possible states of a system.; and

D) The number of degrees of freedom of a system is the smallest number of variables needed to fully describe the system. For example, a needle on the ground has three degrees of freedom: Two to describe the position of its tip and one for the angle between the needle and a given reference line.

Mathematicians give each knot a unique algebraic formula. Equivalent nodes have the same formula “signature”. Mathematicians hope that unequal knots have different “signatures”, so that the algebraic formulas categorize all the knots. Just as police categorize suspects by their fingerprints.

An impossible object (also known as an impossible figure or an undecidable figure) is a type of optical illusion that consists of a two-dimensional figure which is instantly and naturally understood as representing a projection of a three-dimensional object but cannot exist as a solid object. Impossible objects are of interest to psychologists, mathematicians and artists without falling entirely into any one discipline.