Paradox

Beyond Constructible Forms: The Undecidable Bars

Gianni Sarcone

In 1934, the Swedish artist Oscar Reutersvärd sketched a peculiar triangle made of small cubes, neatly aligned on an isometric grid. Everything looked geometrically sound—until the mind tried to assemble it in real space.

Soon after, Lionel Penrose and Roger Penrose published their famous Penrose Triangle—three beams joined by apparently right-angled joints—and M. C. Escher explored similar paradoxes in works such as Waterfall and Ascending and Descending.

This triangular paradox has distant echoes in ancient Greek geometry, but Reutersvärd gave it a clear visual form: the impossible figure.

That’s the charm of impossible figures: every part looks right, yet the whole quietly breaks reality.

I began exploring these paradoxical structures in the 1980s. My interest grew naturally from the meeting point of two inclinations: a mathematical curiosity about spatial logic and a visual fascination with form. Over time I produced many variations—sometimes rediscovering ideas that others had already touched upon, occasionally arriving at configurations that felt genuinely new. In geometry, complete novelty is rare; most discoveries emerge as unexpected turns within an existing landscape.

One example from that period is the study shown here, created in the late 1990s and titled Undecidable Bars.

© Gianni A. Sarcone, 1997-2001

Parallel bars appear to run calmly side by side, yet their connections quietly sabotage the logic of space. Perspective slips from one segment to another, forcing the eye to accept incompatible viewpoints at the same time.

Each element seems perfectly normal.
Together, they form a structure that cannot exist.

Some bars appear to pass through others; some join where no joint should be possible. The geometry behaves as if the object were bending through space, while every line still respects the conventions of perspective drawing.

The result is an undecidable figure—a form the eye can follow effortlessly, but the mind cannot reconstruct.

Available as fine art print from my online gallery.

Over the years I have created hundreds of images built on similar principles, across different formats and media. If these works interest you for a book, exhibition, or monograph, feel free to contact me.

IF: The Two-State Threshold

Gianni Sarcone

I’ve always been drawn to impossible objects—those forms that slip between logic and illusion, never fully settling into one or the other.

This piece grew out of an old idea I felt compelled to revisit, almost as if reopening a long-forgotten door. A binary door, in fact—one that leads to two distinct worlds. Depending on how you look at it, it shifts, tilts, and reveals something else.

It starts with pencil on paper. A loose, intuitive phase where the form finds its way. From there, I move into FreeHand MS—an old tool I’ve never quite let go of. It still gives me a certain precision and feel I can’t replace. Finally, I refine the piece in Photoshop, adjusting, balancing, pushing it toward that delicate point where everything holds together.

There’s still work to be done. Something remains unresolved—but maybe that tension is part of what keeps it alive.

When Water Decides to Defy Gravity

Gianni Sarcone

My minimalist tribute to M. C. Escher: an animated “impossible waterfall,” drawn frame by frame. It’s not exactly my usual artistic language, but I had great fun creating it, and I hope you’ll enjoy watching it as much as I enjoyed making it.

As you can see, the isometric structure links impossible angles to create a continuous water channel that appears to flow upward in a loop, falling from a high point yet seemingly returning to the top.

The “impossible waterfall,” reimagined in a lavish Rococo style, rendered as a surreal illustration for a book project.

Creating a New Impossible Cube: From Concept to Print

Gianni Sarcone

Impossible or undecidable figures have long fascinated artists, mathematicians, and viewers alike. Their appeal lies in a delicate tension: the structure appears perfectly logical at first glance, yet closer inspection reveals spatial contradictions that cannot exist in the physical world. My latest work revisits an idea I first explored in the 1990s—an impossible Rubik’s-style cube—now developed into a new series built across several stages, from hand-drawn construction to digital refinement and photographic interpretation.

The project began with a simple geometric framework—interlocking beams arranged to suggest a stable cubic volume. The challenge was to reinterpret an apparently ordinary three-dimensional cube into an ambiguous form that still appears structurally plausible. Through careful adjustments of line weight, contrast, and directional and formal cues, the cube gradually shifts from perceived solidity to spatial uncertainty, so that as the eye moves across the image, the object quietly reorganizes itself, producing a surreal perception in place of a coherent physical structure.

impossible cube
Here is the original version of the project, refined from my initial hand-drawn construction and carefully reconstructed using FreeHand MX

Two of the final images belong to the Op Art tradition, where sharp black-and-white geometry emphasizes visual tension and rhythmic structure. These compositions highlight the cube’s architectural clarity while allowing the paradox to emerge naturally from the viewer’s perceptual processing. The remaining two images take a different path: they present the object in a photographic setting, rendered with realistic lighting and textures.

impossible cube etched
Astraea Paradox Cube: Available as fine art print.
Rubik’s Paradox Cube: Available as fine art print.

Together, the four images form a small visual narrative—construction, transformation, and illusion—showing how a purely conceptual structure can evolve into multiple aesthetic forms. The Op Art versions focus on perceptual mechanics, while the photographic interpretations suggest how an impossible form might inhabit the physical world, even if only in appearance.

Fine art prints and canvas editions from this series are available through my official gallery shop, where each piece is produced using archival materials designed for long-term display.

Collectors and galleries interested in larger formats or special editions may also contact me directly for availability and production details. This series continues my exploration of perceptual geometry, where simple shapes become instruments for questioning how we construct space, depth, and visual certainty.

Punctum Temporis

Gianni Sarcone

What is an instant—the punctum temporis—that Plato called ἐξαίφνης (exaíphnēs), the sudden? Is it a vanishing point between past and future, or the hinge on which both unfold? Plato saw it as an interruption in the flow of time, a fleeting spark where change occurs, yet which itself seems to escape duration. Augustine later reflected that the present, though indivisible, lives within us as the tension between memory and expectation.

Bergson went further, arguing that real time—la durée—cannot be reduced to a series of measurable instants. If an instant is infinitely small, it cannot be summed; if it can be summed, it is no longer an instant. Thus arises the paradox: if the present is composed of infinite instants, how can it ever be said to exist?

Perhaps time is not made of points but of relations—of movement, perception, and becoming. The instant would then be less a unit of time than a threshold of consciousness, the meeting place of continuity and change. In that sense, punctum temporis is where time reveals its true nature: elusive, dynamic, and inseparable from the act of being.

The Cube That Lies

Gianni Sarcone

I’ve always been drawn to the architecture of geometry. The hexagon, with its quiet strength and symmetry, sits at the root of so many spatial illusions—it’s the seed of cubes, isometric grids, and 3D paradoxes. From this shape, I began exploring structures that bend logic and perception, eventually giving life to a trio of optical works: Enigma 1, Enigma 2, and Enigma 3.

enigma 1
Enigma 1 – Fine Art Prints & T-shirts.
enigma 2
Enigma 2 – Fine Art Prints & T-shirts.
enigma 2
Enigma 3 – Fine Art Prints & T-shirts.

Each piece is built around the visual tension of the impossible cube, created by merging two tribars in perfect isometric perspective. The lines suggest solidity, yet the form escapes reality—what looks structurally sound unravels the moment the eye tries to make sense of it. That’s the game I love to play: where geometry behaves, but perception rebels.

These “Enigmas” are spatial riddles dressed in stripes and angles, each one twisting the viewer’s reading of depth, volume, and continuity in its own way.

Enigma 1 – Prints & T-shirts.
Enigma 2 – Prints & T-shirts.
Enigma 3 – Prints & T-shirts.

Relative Size Illusions

Gianni Sarcone

Here are two relative size illusions I described back in 1997 and 2013.

The first, called Sarcone’s Crosses, challenges classic illusions like the Ebbinghaus illusion (Titchener Circles, 1898) and the Obonai square illusion (1954). It features a cross (the test shape) surrounded by squares of different sizes.

As shown in Fig. 1.a, 1.b, and 1.c, the three blue crosses are all the same size — yet the one on the left (Fig. 1.a) appears larger. Surprisingly, the illusion still works even when smaller squares completely cover the cross (Fig. 1.c).

So, the size of surrounding shapes doesn’t always dictate how we perceive the central one.

In the second illusion (Fig. 2.a and 2.b), due to assimilation, the red diagonal inside the larger ellipse seems longer — but the blue line is actually the longest.

Perception loves to play tricks on us.

sarcone's relative sizze illusions

You can explore more of my illusions and visual inventions on my official site: giannisarcone.com

Smelling the Color 9: When Numbers Take Shape and Color

Gianni Sarcone

In English, the expression to smell the color 9 describes something completely impossible…

And yet, some people have the unusual ability to mentally visualize colors or spatial patterns when thinking about units of time—or more broadly, numbers. This phenomenon, known as synesthesia (from the Greek syn, “together”, and aisthēsis, “sensation”), occurs when stimulation of one sense involuntarily triggers sensations in another. It’s not a figure of speech—these perceptions feel very real to those who experience them.

The first documented case in medical literature appeared in 1710. Dr. John Thomas Woolhouse (1650–1734), an ophthalmologist to King James II of England, reported a blind young man who claimed he could perceive colors induced by sounds.

Neuroscientist Vilayanur S. Ramachandran and his team at the University of California, San Diego, observed that the most common form of synesthesia links “graphemes“—letters or numbers—to specific colors. Since my work bridges art and mathematics, I’ll focus here on number-based synesthesia.

People who experience synesthesia in its pure form are relatively few. However, many report strikingly similar associations between numbers and colors or spatial layouts, suggesting these perceptions aren’t just products of imagination or attention-seeking. For example, number–form synesthesia may result from cross-activation between brain regions in the parietal lobe that handle numerical and spatial processing. In contrast, number–color synesthesia likely stems from an overabundance of connections between adjacent areas that interfere with each other when triggered (see fig. 1 below).

brain synesthesia

Figures 2 and 3 illustrate common synesthetic patterns—either as color associations (fig. 2) or spatial arrangements (fig. 3, based on observations by Sir Francis Galton). Statistically, people often associate the digits 0 or 1—and sometimes 8 or 9—with black or white. Yellow, red, and blue are typically linked to smaller digits like 2, 3, or 4, while brown, purple, and gray tend to be tied to larger ones like 6, 7, or 8. Curiously, it’s not the idea of the number but the visual form of the digit that seems to trigger the sensation. For instance, when the number 5 is shown as the Roman numeral V, many synesthetes report no color at all.

color number synesthesia

And you—do you see numbers in color or arranged in space? Feel free to share your synesthetic experiences with me.

Misdirection → Illusion → Aha! Moment…

Gianni Sarcone

How misdirection, illusion, and wonder shape my creative process.

The path from misdirection to revelation is at the heart of how illusion and wonder spark insight. Misdirection steers our attention—often subtly—away from what truly matters. It disrupts our expectations, creating a gap between what we see and what is. Within that gap lies the illusion: a crafted discrepancy, a visual or cognitive sleight-of-hand that unsettles our perception.

But the magic doesn’t end there. When the illusion is cracked—when the mind shifts, recalibrates, and sees—the famous Aha! moment erupts. That flash of understanding isn’t just delightful; it’s deeply educational. It rewires how we interpret the world.

This sequence—misdirection, illusion, revelation—mirrors the creative process itself. It shows how confusion, when carefully designed, can be a gateway to clarity. In the right hands, illusion is not deception—it’s a tool to awaken curiosity, stretch perception, and provoke insight. Wonder, in this sense, becomes a powerful cognitive catalyst.

That’s why my art and, I believe, my writing, revolve around this sense of wonder—arguably the most direct and playful route to that pleasurable, often conflicting moment of insight: the sudden discovery of something previously unknown.

Hands-On Wonders: A Mathemagical Collection

Gianni Sarcone

Ever wondered what happens when math puts on a magician’s hat? These books are the distilled magic of my hands-on math workshops across Europe — from Paris to Palermo, Geneva to Ghent — where paper folded, minds twisted, and logic sparkled in English, French, and Italian!

Impossible Folding Puzzles

1) “Impossible Folding Puzzles and Other Mathematical Paradoxes” — a playful dive into mind-bending problems where nothing is quite what it seems. Can a puzzle have no solution… or too many? Dare to fold your brain.

Still available on Amazon.

2) “Pliages, découpages et magie : Manuel de prestidi-géométrie” — where math meets illusion to spark curiosity and creativity.
Perfect for teachers, students, and curious minds: touch, fold, cut… and let the magic unfold!
Available on Amazon.

2) “Pliages, découpages et magie : Manuel de prestidi-géométrie” — un livre où maths et illusion se rencontrent pour éveiller curiosité et créativité.
Pour enseignants, élèves et esprits joueurs : touchez, pliez, découpez… la magie opère!
Dispo sur Amazon.

Pliage decoupages

3) “MateMagica” —  They say there’s enough carbon in the human body to make 900 pencils… but just one is all you need for these clever puzzles!
Fun, surprising, and thought-provoking — because, as Martin Gardner put it, “Mathematics is just the solution of a puzzle.”
Now on Amazon.

3) “MateMagica” —  Si dice che nel corpo umano ci sia abbastanza carbonio per 900 matite… ma per questi rompicapi ne basta una!
Sorprendenti, divertenti e stimolanti — perché, come diceva Martin Gardner, “la matematica è nient’altro che la soluzione di un rompicapo.”
Disponibile su Amazon.

I write and illustrate my own books in five languages: English, French, Italian, German, and Spanish.
If you’re a publisher or literary agent seeking original, high-quality educational content that blends creativity with clarity, I’d be pleased to explore potential collaborations.