Beyond Constructible Forms: The Undecidable Bars

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.

The Logic of Abstraction

Some say abstract art is non-representational—that it avoids visual reality and relies on color, shape, form, and gesture to trigger emotion or thought. I see it differently. Abstraction does not reject reality; it reframes it. It is a change of optics, not a disappearance of the world.

Take, for example, this video of goldfinches perched on swaying thistles. At first glance, does it not resemble an abstract painting? Rhythms, repetitions, subtle chromatic tensions, forms dissolving into movement. From there, one could push the abstraction further with the slightest shifts in shape or color—without betraying reality, only rephrasing it.

This idea is hardly new. From Cézanne’s insistence on treating nature through cylinder, sphere, and cone, to Kandinsky’s claim that abstraction reveals inner necessity rather than surface likeness, many artists and thinkers have argued that abstraction sharpens perception instead of diluting it. Even in cognitive science, perception is understood as an active construction, not a passive recording of facts.

Abstraction starts precisely there: with attention. Not with denial, not with decoration, but with the recognition that reality is already structured, already abstract, long before the artist intervenes.

Seeing Beyond Lines: The Illusion of Form

We usually perceive lines and shapes as forming a figure, while the paper and surrounding white space recede into the background. Yet, under certain conditions, what we assume to be background can itself acquire form and meaning. In the illustrations shown here, a clear geometric figure emerges—even though no lines actually define it. These visual phenomena are known as illusory figures.

illusory spring
What kind of 3-dimensional shape do you see?

Illusory figures depend, in part, on the presence of regular gaps within a visual arrangement. When such gaps occur, the visual system instinctively tries to resolve them into coherent forms. These gaps can be created by simple elements, such as circles. When solid black circles and partially “chipped” ones are arranged carefully, they produce striking illusory shapes.

The most familiar example is the Kanizsa triangle. Here, an illusory contour is perceived when black disks with wedge-shaped sections removed are aligned so that their edges define a triangular form in the negative space. Remarkably, this illusory region appears brighter than the surrounding page, even though its physical luminance is identical.

Kanisza triangle
Kanizsa triangle

Etherial Cross

A pattern of black dots forms a ghostly ‘X’ shape through negative space. The arrangement of dots creates a subtle sense of motion and depth. The piece combines the Ouchi effect—where contrast between figure and background generates apparent movement—with the Kanizsa illusion, where the mind completes shapes that aren’t actually drawn.

Sarcone's Illusory Cross
Fine art prints of this Op Art piece are available for purchase through my official gallery.

The effect becomes even more dramatic when the illusion is animated. Rotate the cross, for example, and the X-shaped form—with no explicit outline—appears to emerge from a rigid grid of black dots. The shape is defined solely by subtle local distortions: small asymmetric intrusions along the contours of the X disrupt the regularity of the dots, making the form pop into perception despite having no actual boundaries.

Animated Sarcone's Illusory Cross

Etherial Circle

Using the same technique, we can replace the X with a large O. Now the regular arrangement of black circles is disrupted by a ghostly central shape that seems to lift off the background, almost floating. As above, this Op Art piece combines the Ouchi effect—where the contrast between figure and background creates apparent motion—with the Kanizsa effect, where the mind completes shapes that aren’t actually drawn.

Floating O
Fine art prints of this Op Art piece are available for purchase through my official gallery.

It is also interesting to add a rotational motion to this Op Art piece. I experimented with different solutions, and this one is the simplest yet striking nonetheless.

To conclude this journey into the world of contour illusions, still using black disks as our starting point, here are a few further experiments. They explore negative superpositions, translucent effects, and the emergence of more complex forms—showing how simple elements can give rise to unexpected visual structures.

hidden beauty
Fine art prints of this Op Art piece are available for purchase through my official gallery.

Ananke’s Die

A simple study in visual perception—an exploration of how a plain hexagon can evolve into the illusion of a cube. Through precise geometry and controlled form blending, static lines awaken into rhythm and volume, giving rise to a subtle sense of depth and movement.

Constructing the Illusion

Fig. A — The Base Shape
Start with a regular hexagon. Divide it into three equal diamond shapes (rhombuses)—these represent the three visible faces of the cube. Each diamond has four equal sides: two acute angles (60°) and two obtuse angles (120°). Together, they form the geometric foundation of the cube.

Fig. B — Building Volume with Shape Blends
In Illustrator, or any other vector software, use the Blend Tool to create a shape blend inside each diamond. Start with a small central circle and blend it toward the outer edge of the diamond. Adjust the number of blend steps to control how smooth or tight the transition appears. This process builds the cube’s apparent volume and visual tension. You’ll notice that the distance from corner to corner in the nested, diamond-like shapes is slightly greater than from side to side, creating subtle gaps that lead the eye to perceive an X across the surface.

Fig. C — Perspective and Transformation
Distort slightly the hexagon to set the three diamonds in perspective. This step transforms the flat figure into a die-like cube, giving it spatial depth and presence.

ananke cube

Enhancing the Optical Effect
Next, add horizontal background lines and some color, as shown in the two examples in the image. You can also adjust the illusion by making the visible faces of the die appear slightly concave, as in the figure on the right. This effect is created by shifting the concentric, nested diamond shapes slightly off-center—the position of the central ellipse determines whether the die appears concave or convex.

 two Ananke dice

Below is the finished stage of the work. Curiously, the cube appears to hover, slide, and even emit a faint blue glow—though it remains entirely black and motionless.
Ananke’s Die is a study I began in 2010, a continuing exploration of how repetitive lines and geometric precision can trick the mind into sensing motion and color where none exist.

Ananke die

You can get Ananke’s Die as a fine art print or canvas, available in different sizes and finishes.
👉 Buy it here

Why Ananke’s Die

I titled this work Ananke’s Die after Ananke, the Greek goddess of necessity and fate.
The cube, a symbol of structure, represents order and control. Yet the three visible faces that seem to define its volume are an illusion—shifting and unstable.
Under the viewer’s gaze, the shape changes, its meaning shifts, yet the form remains.
This illusory die shows the balance between order, perception, and destiny, reminding us that what we think we control often exists within the unpredictable interplay of vision and inevitability.

This image also triggers multiple associations in a loop: hexagon, cube, die, chance, illusion, order, fate, contradiction. These connections show how perception mixes stability and randomness, revealing that what we see is shaped as much by the mind as by reality.

Forbidden Brews: When Coffee, and Chocolate Stirred Trouble

In the 17th century, some of today’s most beloved drinks — coffee, tea, and chocolate — were once viewed with deep suspicion. When these “divine beverages” first arrived in Europe, civil and religious authorities saw them as exotic, even subversive. Their foreign origins and stimulating effects made them objects of fascination, controversy, and at times, prohibition.

Coffee traces its roots to the highlands of Ethiopia, where the Coffea arabica plant produces the precious beans that, once roasted and brewed, yield the dark, fragrant drink we know today. From Ethiopia, coffee spread to Yemen and across the Islamic world before reaching Europe. In 1672, an Armenian merchant named Pascal introduced the first coffee to Paris, setting up a small stand near the Saint-Germain fair — an event that marked the beginning of France’s enduring love affair with the drink.

Chocolate has an equally captivating story. It comes from the seeds of the Theobroma cacao tree, native to Central and South America. The Maya and Aztec peoples prepared it as a sacred, bitter beverage, often mixed with spices and chili — a divine elixir meant to awaken both body and spirit.

These exotic drinks, however, sparked strong reactions. Coffeehouses became lively meeting places where new ideas brewed alongside cups of steaming coffee — much to the concern of kings and clergy. Chocolate, too, stirred debate within the Church: some saw it as sinful indulgence, others as a heavenly pleasure.

Even in the Age of Enlightenment, the story continued to brew with conflict. In 1777, Frederick the Great of Prussia tried to ban coffee altogether. He feared it would replace beer — the national drink and a source of tax revenue — and even claimed that beer made his soldiers strong, while coffee made them weak. To enforce his will, he appointed “coffee sniffers” (see picture) to hunt down those secretly roasting beans.

Despite resistance and regulation, these once-suspect beverages soon became powerful symbols of refinement, curiosity, and creative thought. Today, coffee, tea, and chocolate remain faithful companions of conversation, reflection, and imagination.

"Die Kaffeeriecher" (The Coffee Sniffers). After a painting by L. Katzenstein
“Die Kaffeeriecher” (The Coffee Sniffers, 1892). After a painting by L. Katzenstein.

A Personal Reflection: Forbidden Pleasures and Creative Sparks

This history of suspicion and fascination mirrors the journey of modern art. Like coffee, tea, and chocolate, art has often been feared, censored, or condemned. Impressionists were mocked for daring brushstrokes; avant-garde movements were branded degenerate under totalitarian regimes. But prohibition, ridicule, or misunderstanding never extinguished their appeal — instead, it made them irresistible.

I see in these stories a reflection of human curiosity itself: the desire to taste, to see, to explore what lies beyond the accepted, the safe, the ordinary.

The Cube That Lies

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.

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.

The Story of Blue

Sonne au comble de l'or
l'azur du jeune hiv
er
– Paul Valéry

For me, blue is air, wind, melancholy. It is like a Fellini film—always haunted by the whisper of the wind, the scent of things, and the fleeting moments of time we wish we could hold in an eternal present. Blue can carry regret; it is, in a way, a conservative principle. Blue has always held a strange duality: so immediate, yet once so elusive. Looking back to antiquity, it fascinates me that Homer never described the sea as blue, but as ‘wine-dark.’ In Greek, kyaneos evoked a dark, mineral depth, while glaukos hovered between gray, green, and blue. The Romans spoke of caeruleus, tied to the sky (caelum), and of lividus, the bruised, bluish tone of flesh. They never elevated blue; it was the color of outsiders, the hue of Celts painted in woad. Only the Egyptians seemed to truly revere it. They invented Egyptian blue, the first artificial pigment, and made it the shade of eternity and the divine.

In the Middle Ages, blue nearly vanished from prominence. Christian art turned to red, white, and black, leaving blue to the margins. Yet the word itself was evolving. The Germanic blāo—and its Vulgar Latin adoption, blavus—once meant something far less precise: shimmering, lustrous, dark, gray, even pale or yellowish. That ambiguity makes me realize how fluid color once was, before language fixed its boundaries. From blāo we inherited English blue and French bleu. Meanwhile, through Arabic lazaward came the words azure, azzurro, and azul, all born from lapis lazuli.

Then, in the 12th century, blue was transformed. Artists began clothing the Virgin Mary in ultramarine, ground from lapis lazuli more costly than gold. What had been humble became heavenly. Heraldry embraced azure as one of its noble tinctures, and kings like Louis IX made blue their emblem. A forgotten hue became a sacred, regal presence.

By the Renaissance, ultramarine shone as a color of prestige, truth, and constancy. Painters reserved it for the highest subjects, poets linked it to loyalty, and explorers carried its name across the seas as azul. Blue had finally claimed its place.

The modern era democratized it. Prussian blue appeared in 1704, followed by cobalt, cerulean, and synthetic ultramarine. Blue poured into uniforms, flags, and revolutions; it became the shade of liberty, of nations, of collective identity. Goethe called it spiritual, a color that retreats yet draws the soul inward.

In the 20th century, painters gave blue an entirely new destiny. The Expressionists used blue to conjure emotion, depth, and inner turbulence—think of Kandinsky, who saw blue as moving ‘toward the infinite,’ or Franz Marc, who painted blue animals to symbolize spirituality and hope. Even Van Gogh’s Starry Night swirls with blue, capturing both wonder and melancholy. Later, Picasso entered his Blue Period, transmuting sorrow into tone. Yves Klein went further still, reducing blue to its purest intensity with his International Klein Blue, turning it into an immaterial field of experience. For Klein, blue was not simply a color but ‘the most abstract color of all,’ a space in which one could lose oneself.

Today, I see blue everywhere. In the jeans that became the fabric of daily life. In corporate logos designed to inspire trust. In flags that mark belonging. Blue is calm, yet melancholy; humble, yet exalted. Once overlooked, it is now the world’s favorite color.

Blue is more than a hue. It is an idea shaped across centuries—etymologically born of caelum, blāo, and blavus; symbolically stretched between heaven and earth; culturally tied to faith, power, and identity. Its story mirrors our own: restless, mutable, and ever searching for meaning.

I’ll end with a little story from my own family. After World War II, my paternal grandfather, a house painter, bought a massive surplus of blue paint from the military. He had so much of it that whenever a client asked how he planned to paint their home, his answer was always the same, year after year: “All in blue, yes, all in blue…” (in our dialect, tutt’azzurr, tutt’azzurr).

That simple phrase stuck. Before long, the whole village started calling our family the Tuttazzurr family, and I myself became known as Giuan Tuttazzurr. Blue marked our home, but also my identity and my story—etched in memory and in name.

Moonlight Reflections on the Waters

A memory from Japan, where I lived briefly in the 1980s. This piece recalls earthy colors, organic shapes, and fragments of that time. A circle emerges from a flowing field of triangles—like ripples of moonlight dancing on the sea near Kamakura.

Immersing in my Op Art is entering a space where opposing forces meet, interlock, and balance with precision and intensity. Each piece is a silent dialogue of form, line, color, perception, and mathematical structure, anchored in the language of symbols. Beneath the surface, it engages archetypes and ancient rites that still resonate in the collective unconscious.

This unique op art piece is available as fine art prints and canvases in my online gallery.

Rediscovering Flutex: Simple Glass, Complex Illusions

I’ve been toying with the idea of revisiting an old, low-key material for my art: Flutex.

If you haven’t heard of it, Flutex is a patterned industrial glass from the 1930s and ’40s, mostly used to give a bit of privacy in bathrooms and office partitions.

In the ’70s, Op artist Sydney Cash started playing with this glass and found that its ribbed surface works like a lenticular screen—showing different images depending on how you look at it. The effect? Hypnotic, shifting artworks that change as you move around them.

It’s just simple glass, but it tricks perception in a really cool way.

I’m seriously considering giving it a try myself—there’s something about that mix of humble material and complex visual play that feels worth exploring again.