Tag: mathematics

The Golden Ratio: Nature’s Perfect Proportion The Golden Ratio in Nature The Golden Ratio in Art and Design A Number That Never Ends The golden ratio, represented by the Greek letter φ (phi), is approximately equal to: φ ≈ 1.61803… It is defined as a ratio where the ratio of the read more

The Riemann Series Theorem The Key Idea Why Does This Matter? The Riemann Series Theorem The Riemann series theorem, named for and rigorously proved by German mathematician Bernhard Riemann, involves conditionally convergent series. A conditionally convergent series is one that converges only under the condition that the sign of each term read more

Fractals introduced the idea of non-integer dimensions, also called fractional dimensions. These are used to measure the complexity of objects that display self-similarity, meaning patterns that repeat at different scales. The Koch Snowflake Mandelbrot and Fractal Geometry Where Do Fractional Dimensions Appear? The Koch Snowflake A famous example is the Koch read more

Khinchin’s constant, denoted by K, was proven by the Russian mathematician Alexander Khinchin in 1934. Definition Open Questions Why Is This So Remarkable? Definition Let x be a real number with a continued fraction expansion where a₀ is an integer and a₁, a₂, a₃, … are positive integers (the partial denominators). read more

Let’s start by drawing a circle named C₁. Draw a second circle, C₂, that touches C₁ at just one point. Now draw a third circle, C₃, that is tangent to both C₁ and C₂. With these three circles in place, we can always draw exactly two more circles that are tangent to all three. Let’s read more

Imagine a game beginning with a stake of $2. The stake is the amount the player will be paid at the end. The player flips a coin, and if it lands on tails, the stake doubles. Otherwise, the game ends and the player collects the stake. Calculating the Expected Value The read more