The Wolfram Alpha Logo

 I used Wolfram Alpha today to determine when I would be 77.77777 years old. 

Yep, 9:21pm on January 11th 2027, midway between my father's birthday on January 9th and my brother Tony's birthday on January 13th. More about that later but, perhaps because of my recent foray into sketching, I happened to notice the logo shown below that appears on the web page. Oddly, I failed to notice the red stellated polyhedra that appears in the image above, maybe because it's quite small.

I asked Gemini the following:

What can you tell me about the Wolfram Alpha logo. It’s an interesting shape. Does it have any history or special significance? Why was this shape shown? Does it have name?

This was the response:

The shape shown is a 2D stylized projection of a \( \textbf{rhombic hexecontahedron}\).

Mathematical History and Properties

• Origins: The shape was formally described mathematically in 1940 by the German mathematician Helmut Unkelbach.

• Geometric Structure: It is a \( \textbf{non-convex zonohedron} \) composed entirely of 60 intersecting \( \textbf{golden rhombi} \) (rhombuses whose diagonals align exactly with the golden ratio).

• Vertices and Edges: The structure features 62 vertices and 120 edges, all organized with perfect \( \textbf{icosahedral symmetry} \).

• Stellation: It is formed as a stellation of the \( \textbf{rhombic triacontahedron} \), meaning it is created by extending the face planes of a core polyhedron until they intersect again to form a new, more complex outer boundary.

• Internal Geometry: The internal volume of the shape can be cleanly dissected into 20 acute \( \textbf{golden rhombohedra} \) that meet exactly at the center point.

Significance and Selection as a Logo

• The "Spikey" Tradition: The rhombic hexecontahedron continues a decades-long corporate tradition initiated by Stephen Wolfram. Since the release of Mathematica 1.0 in 1988, Wolfram products have been represented by complex, mathematically generated 3D polyhedra, affectionately known internally as "Spikeys." While early Mathematica logos featured shapes like the stellated icosahedron and hyperbolic dodecahedron, the rhombic hexecontahedron was designated specifically for the launch of the Wolfram|Alpha engine in 2009.

• Visualizing Computation: The logo was selected to reflect the computational complexity, algorithmic precision, and mathematical rigor underpinning Wolfram|Alpha. The shape’s composition—made of identical, mathematically strict intersecting faces that combine to form a highly intricate and globally symmetric structure—serves as a visual metaphor for the engine's ability to process massive amounts of disparate raw data and compute unified, structured answers.

Here is what Gemini did with a sketch that I made of the shape using the following prompt:

This image is a 2D stylized projection of a rhombic hexecontahedron, used by Wolfram Alpha as its logo. I need you impose a strict symmetry on the shape and make it look as solid and 3D-looking as possible with shading or whatever. The faces I’ve represented as windows and these should be rendered as translucent with a yellow light emanating, seemingly from within. The overall shape should then look like a polyhedral house with a warm and inviting glow.

Instagram Link

The Golden Rhombus and Golden Rhombohedron referred to earlier look this:

A rhombohedron shown above is an acute golden rhombohedron and it is a cube that has been pushed out of shape so that all of its faces are rhombi. A golden rhombus has a ratio of long diagonal to short diagonal of \( \phi \; : \; 1\). A golden rhombohedron is simply a rhombus with all faces golden rhombi. 

• The acute angle in a golden rhombus is given by \( 2 \text{arctan} (1/ \phi) \approx 63.43^{\circ} \)
• The obtuse angle is given by \( 2 \text{arctan} ( \phi) \approx 116.57^{\circ} \) 

Twenty acute golden rhombohedra can be combined to form a solid rhombic hexecontahedron. There is also an obtuse golden rhombohedron but I'll deal with that later.