Amazingly, this diagram presents 4 distinct representations that convey the same value. These include a percentage (75%), a fractional expression (3/4), a decimal notation (0.75), and a visual representation (3 out of 4 blue boxes filled).

### Geometric Art Mobile

The Magic of Projective Geometry: Installation by Johannes Langkamp (German/Dutch, born 1985).

### Aperiodic Tiling

An aperiodic monotile, humorously dubbed an ‘einstein’ (from the German term “einstein,” meaning “one stone” or “one tile”), is a single tile that covers a surface without repeating patterns. This posed a challenging question for some time: could such a tile exist, or was it impossible?

In 2023, David Smith and his team provided an answer. They discovered a simple tile called a “hat” that can achieve this aperiodic tiling. The geometric shape of the “hat” tile is based on the symmetry and edges of a hexagon, as shown in the picture. According to Smith, this tile, along with its reflection (shown in blue), enables an array of unique, non-repeating tile arrangements. The “hat” falls within the broader category of Smith–Myers–Kaplan–Goodman-Strauss tiles.

### Early Mathematics

## Numeral systems began as simple tally marks, created by piercing holes or engraving lines on a material to represent and count units.

With the development of agriculture came the need for practical tools for surveying cultivable plots, calculating seasonal phenomena, and levying fair taxation. Consequently, numeral systems emerged within great civilizations, such as the Sumerian and Babylonian cultures (Mesopotamia, ancient Iraq), and ancient Egypt.

These are the reasons why the Ishango bone artifacts represent such an astonishing discovery and serve as evidence that humans employed the concept of numbers in ancient times. Unearthed in ancient Belgian Congo (present-day Democratic Republic of Congo), these archaeological finds take the form of bone sticks, roughly ten centimeters in length, affixed with a piece of quartz at one end. They are thought to be the earliest known pocket calculator, dating from 20,000 to 35,000 years ago.

Continue reading “Early Mathematics”### Unlocking the Fraction

Use a trick to quickly and effortlessly determine the fraction that falls between 3/4 and 4/5.

### Set Theory ⋃ Humor

Practical examples of intersection and union…

### Averaging square roots

Is the average of two square roots less than, equal to, or greater than the square root of the average?

### Visual Math (Trisection)

Explore five intriguing methods of segment trisection.

### Lateral Mathematics

High math skills are required to solve this puzzle…

Fill in the three boxes below using any of the following numbers: 1, 3, 5, 7, 9, 11, 13, 15.

You are allowed to repeat the numbers.

### The Geometry of the Bees

When constructing a honeycomb, bees aim to minimize wax usage and honey consumption, using the least wax necessary for a comb with maximum honey storage. The wax cells are designed with interlocking opposing layers, sharing facets at closed ends while having open ends facing outwards (see fig. 1). Each cell is a ten-sided structure with a rhombic decahedron form – a hexagonal prism with three rhombi at its closed end (fig. 2). Mathematicians have extensively studied the highly efficient isoperimetric properties of these cells. The question remains: What angle alpha maximizes volume while minimizing surface area on each cell face when the hexagonal prism’s faces have a width of 1 unit?

Continue reading “The Geometry of the Bees”