Geometry

The Geometry of Wonder

Gianni Sarcone
impossible objects: nested tribars

One of my finest sculptures: nested triangles, the central one in wood, the others in stainless steel…

nested tribars

Just kidding. It’s an impossible tribar composition — a sculpture that can exist only in the eye and the mind.

The Penrose triangle (or Penrose tribar) is generally credited to Roger Penrose, who, together with his father Lionel Penrose, published it in 1958 as an “impossible object.” However, the visual idea of creating paradoxical triangular forms and impossible geometries has older precedents.

Japanese visual culture is often mentioned in discussions of geometric illusion because Edo-period (1603–1868) art, decorative motifs, and craft traditions reveal a deep fascination with complex patterns, ambiguous spaces, and unconventional perspectives. Elements found in traditions such as sankaku mokkō (triangular motifs), karakusa patterns, architectural ornamentation, and woodworking designs show how Japanese artists and craftsmen explored visual structures that challenge perception. While no confirmed direct predecessor of the Penrose tribar has been identified in Japanese art, these examples belong to a broader historical tradition of creating forms that play with geometry and the limits of visual interpretation.

Japanese tribar

A glimpse below into traditional Japanese fabric patterns, where triangular designs reveal how interlaced geometry can create the sensation of impossible structures.

Kagome patterns

Triple Klein Bottle

Gianni Sarcone

Τριαλάβαστρον (Trialábastron), a triple Klein bottle, is an artistic exploration of Klein bottle geometry. In topology, the Klein bottle is a non-orientable surface that cannot be embedded in three-dimensional Euclidean space without self-intersection, though it can be smoothly realized in four dimensions.

A close relative is the Möbius strip, another non-orientable surface in three-dimensional space. On such surfaces, a continuous path can return to its origin with reversed orientation, revealing how geometry can twist the notion of inside and outside.

a triple Klein bottle

Here is another representation of a triple Klein bottle—UK sculptor Alan Bennett’s striking construction of three Klein bottles nested within one another. Of course, “inside” only applies in the sense of their embedding in physical space; from a topological standpoint, they remain entirely separate, non-interacting surfaces.

recursive Klein bottle

Your Brain vs. Geometry: Rooftop illusion

Gianni Sarcone

Believe it or not, the green and purple rooftops are congruent—identical in shape, size, and angles. What changes is not the geometry, but your perception of it. Perspective, orientation, and contextual cues lead your visual system to interpret the same form as two different structures.

rooftop illusion, related to shepard tables

This effect is closely related to Shepard’s tabletop illusion and earlier studies in which identical parallelograms (A and B), when rotated (and superimposed), are perceived as different shapes, as the brain prioritizes interpretation over measurement.

Since some still insist these roofs are not congruent, I’ve put together a short animation that shows otherwise.

animated rooftop illusion

The Borromean Tribar

Gianni Sarcone

This form belongs to the family of impossible figures, more specifically to the Penrose triangle, or tribar.

impossible figure

At first glance, it looks entirely manufacturable. The structure appears to be made from three identical square rings (Fig. 1), each pierced by a circular opening and arranged in three mutually perpendicular planes (Fig. 2). Together, they seem to interlock seamlessly, forming an impossible three-dimensional tribar that evokes a trefoil knot and its endless over-under weaving.

If this sculpture looks physically realizable to you, congratulations: your visual system has just accepted a geometric impossibility.

the making of the borromean tribar

Could such an object be built? Not exactly. One solution would be to twist one of the junctions slightly, but that trick would be immediately noticeable here because some of the angles would no longer appear perfectly square (90°), as shown in the picture below. Another possibility would be to leave a small gap at one junction and view the sculpture from a carefully chosen angle, allowing perspective to hide the discontinuity.

This is how a real 3D Borromean Tribar would appear…

How a Human Bone Inspired the Eiffel Tower

Gianni Sarcone

Few people know that the human femur—the body’s largest and strongest bone—played an indirect role in the thinking behind the design of the Eiffel Tower.

Part of the tower’s structural logic can be traced to Swiss engineer Maurice Koechlin, chief engineer in the firm of Gustave Eiffel. While determining how forces would travel through the iron frame, Koechlin applied a principle that places material along the natural paths of tension and compression.

A comparable pattern had been described earlier by Zurich anatomist Hermann von Meyer. His research revealed that the femur’s internal structure forms a network of delicate struts known as “trabeculae.” These tiny elements follow the directions of mechanical stress inside the bone, creating a highly efficient system of support—even though the femoral head sits off-center from the shaft.

The mathematician Karl Culmann later showed that these trabecular patterns correspond closely to the principal stress lines calculated in engineering. His method, called graphic statics, provided a visual way to map how forces move through structures.

This link between anatomy and engineering influenced nineteenth-century structural thinking. The same principle—placing material only where forces demand it—guided the development of lighter, more efficient frameworks in bridges, cranes, and reinforced-concrete designs.

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.

Here’s a simple 18-frame animation of my Op Art piece—work in progress.

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.

Kinegram Exhibits

Gianni Sarcone

Here are two Kinegram installations I made, designed to educate, engage, and spark curiosity in visitors of all ages. These works make motion appear from static forms, offering an experience that is both playful and thought-provoking.

Kinegrams reveal movement through a sliding transparent panel printed with vertical black lines. As the panel moves over the underlying image, hidden sequences appear, animating the drawings like frames of a film. Each interaction lets the viewer explore how motion can emerge from stillness.

The concept of movement from static forms has long interested scientists and philosophers. Movement is a dimension unfolding in space and time—without time, there is no motion. Kinegrams make this idea tangible through touch and visual perception.

These exhibits are simple to set up but produce surprising results. I made the first panel for UNIFI, the University of Florence, and the second for the Mind Games Art Alive Museum in Sydney. Both projects engage and surprise visitors, combining education with visual impact.

Kinegram - UNIFI

“Twisting Cords,” Kinegram exhibit for UNIFI, Florence
As the transparent panel etched with black lines glides across the design, colorful cords seem to twist, winding and unwinding in a mesmerizing, living rhythm.

Flying Birds - Migration

“Flying Birds – Migration,” Kinegram exhibit for Mind Games, Sydney
As the transparent panel etched with black lines glides across the static design in the background, the flock of birds rises and takes wing, transforming a still pattern into living, rhythmic motion.

To see more or discuss a Kinegram installation, visit: https://www.giannisarcone.com/Kinegrams.html