A neat animated tribar! It’s worth noting that the tribar, or Penrose triangle (aka Reutersvärd triangle), attributed to British mathematician Roger Penrose, was not technically ‘invented’ or ‘discovered’ by him. The geometric principles underlying its existence were already evident in Greek and Arabic ornamentation, including tiling and friezes…

### Numbers Defying Ceiling and Floor Functions

Two sequences that agree for an embarrassingly long time.

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### Circles & Roots

Delve into the realm of Sacred Geometry, where circles unveil the elegance of successive square roots from 1 to 6. Extend your exploration with the enigmatic charm of the square root of *Phi*.

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### Apparent size ≠ Real size

Simple demonstration of apparent size and distance… See how the color rings (*annuli*, in mathematical language) fit snugly.

### The Origin of Modern Alphabets

The **Phoenician alphabet** is a writing system exclusively representing consonants, requiring readers to infer vowel sounds. Beginning in the ninth century BC, adaptations of this alphabet thrived, including Greek, Old Italic, and Anatolian scripts. Its appealing feature was its phonetic nature, with each sound (including vowels) represented by a single symbol, simplifying learning to only a few dozen symbols.

### Nested Penrose Triangles

This is an illusory geometric structure that cannot exist in our 3D world. Let’s Explore its captivating depths and intrigue.

Here’s how to create this impossible structure. Start by drawing two parallel lines spaced apart from each other and divide them into 7 equally spaced lines.

Then follow the visual steps A, B, C, and D illustrated below. At the beginning (fig. A), you will need to replicate the alignment of the 9 parallel lines three times while applying a 60-degree rotation to each one, finally arranging them to form a triangle. Subsequently, follow the visual directions in B and C to obtain the figure shown in fig. D.

At last, you can add color and gradients to the structure as illustrated below.

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### Topological Oddity: A Picture-Hanging Puzzle

Imagine the linear pattern as a hanging rope. Now, removing any one of these four nails will cause the entire rope to fall.

### Illusory Structures

That’s what happens when you fall down a Penrose staircase…

### Logarithmic and Fibonacci Spirals in Plant Phyllotaxis

Nature, particularly in plants, features logarithmic and Fibonacci spirals, exemplifying the elegance of natural design and the rhythmic dance of life, encompassing symmetry and other intriguing mathematical phenomena, including recursive functions.

Spiral patterns in plants emerge from their repetitive growth, where each turn closely mirrors the previous one with scaling or rotational adjustments. This growth process, common in nature and known as phyllotaxis, utilizes recursive functions, which can generate logarithmic and Fibonacci spiral patterns.

### Sudoku for Dummies

The binary edition for those affected by number blindness.