Perplexed by AI

 The last post, "Flow of time and flow of space", ended with "In the next post we take a look at what kind of life can reside on the two  infinity circles?".  That was my plan. Trying to bring it to life I started calculating and researching. One thing was leading unavoidably to another, while I was intensively using AI (mainly Perplexity) for digging into "what has been already published on the subject?" My search was mostly unsuccessful. Perplexity could not help itself but to start hallucinating. It was referring me to sources, papers, textbooks, online resources, that did not contain the stuff it was claiming to have. So I was changing my prompts until finally I was led to a paper that made me very very happy. It was like discovering a wonderful meadow with lots of daisies and grazing unicorns. The title of the paper is:   "A primer on the (2 + 1) Einstein universe".

A primer on the (2 + 1) Einstein universe

It was exactly what I needed. The paper is some fifty pages long, and I will be studying it slowly while at the same time describing what I have already learned here. After that we will return to the point where we have left, but with a much broader perspective. The paper is written by five mathematicians, T. Barbot, V. Charette, T. Drumm, M. Goldmann, K. Melnick, while they were visiting the Schrodinger Institute in Vienna. You can download it from arxiv here. It was published in print in"Recent Developments in Pseudo-Riemannian Geometry", (ESI Lectures in Mathematics and Physics), Dmitri V. Alekseevsky and Helga Baum eds.,  EMS - Publishing House, Zürich 2008, pp. 179-229.

Keywords: Minkowski space, spacetime, Lorentzian manifold, conformal structure, Lie algebra, symplectic vector space
"(...) The purpose of this article is to develop the synthetic geometry of the Einstein universe in terms of its homogeneous submanifolds and causal structure..."

I will start with the math in the next post. This post is just an announcement. But I also want to tell you that my view on Artificial Intelligence has evolved. The AI is not "intelligent" at all. It is a useful companion, it has access to huge databases, but occasionally it is simply dumb! It can be also dangerous when you try to rely on it.  Just yesterday I asked Perplexity to sketch a proof of a certain result quoted in the paper I am advertising, but without providing a proof. Perplexity happily "proved" the statement and suggested a reference to look at. I looked and there was nothing even roughly similar there. Then I thought for a while, and decided to ask if another statement, contradicting the first one is true. The answer was: yes, the second statement is true, which means that the first one, the "proven" one, is false. Sure, both statements look "similar", but we are dealing with mathematics. The statements x=y and x≠y look similar! But, while using two-valued logic, only one of them can be true at a time.

Here is another example. I asked Perplexity to analyze the picture below and describe what kind of optical illusion it provides:

Here is the reply:

The attached picture shows a classic impossible object illusion, specifically related to the genre of the Reutersvärd Triangle or Penrose Triangle.

In the image, you see a group of cubes arranged in a triangular configuration on an isometric grid. At first glance, each cube's shape appears normal, but as you mentally connect the arrangement as a 3D object, you realize the structure can't exist in real 3D space: the cubes fit together in a way where the geometry of connections between them is self-contradictory.

This is an example of an impossible figure illusion: the brain tries to interpret the arrangement as a three-dimensional object, but the spatial relationships between the cubes force contradictory perspectives. The drawing leverages local plausibility with global impossibility—meaning each small part looks realistic, but the whole configuration violates Euclidean geometry when viewed as a solid object.

Such illusions are used to illustrate how perception can be tricked by carefully designed two-dimensional images that depict impossible three-dimensional shapes—one of the pioneering styles introduced by Oscar Reutersvärd and later made famous by Penrose and Escher.

Well, I do not see anything "impossible" in the picture!
So I asked the same question Yandex AI, Alice. The answer was much much more satisfactory:

The optical illusion in the attached picture is an example of a 3D isometric illusion (also known as a multistable perception illusion or illusion of depth).

Explanation:

The image shows a group of cubes drawn on a grid using isometric perspective — a technique where parallel lines remain parallel, creating a 3D effect on a 2D surface.

The shading and positioning of the cubes trick the brain into perceiving depth and volume, even though the drawing is flat.

The illusion works by:

Using consistent angles (typically 30° or 45° from horizontal) to represent the three dimensions (length, width, height).

Applying shading to simulate light and shadow, which enhances the 3D effect.

Arranging the cubes in a way that mimics spatial relationships (some appear in front, others behind).

Key characteristics:

Ambiguity: The viewer’s brain struggles to reconcile the 2D lines with the 3D interpretation.

Forced perspective: The grid lines help maintain the illusion by suggesting a consistent spatial grid.

Depth perception: The shading and cube arrangement create a convincing sense of depth.

Similar illusions: This type of illusion is related to:

Necker cube: A classic example of a reversible 3D cube drawing.

Impossible objects: Like the Penrose triangle, which uses similar isometric techniques but creates geometrically impossible shapes.

Conclusion: This is a classic example of how 2D drawings can deceive the visual system into perceiving three-dimensional space, relying on perspective, shading, and spatial arrangement.

For a perplexed Reader here is the link to  Oscar Reutersvärd's "impossible art"

Afternotes

10-11-25 19:12 I have decided to move from Blogger to Substack completely. Here is the Substack version of this post.

11-11-25 11:32 Or, alternatively, I will move to my very old blog, where I used to have latex installed. It is disfunctional now, but I will try to fix this old installation at arkadiusz-jadczyk.eu.