Changing Planet

LEONARDO’S BRIDGE: Part 3. “Vebjørn Sand and Variations on a Theme by Leonardo”

The Ultimate Fusion of Form and Function. The blue hour photo shows Leonardo’s Bridge inspired by a 500-year old sketch. “One of the five coolest bridges in the world,” declared Wired Magazine (2005). The photo above appeared previously in the author’s book, Leonardo’s Universe (National Geographic Books, 2009)

Vebjørn Sand is a contemporary Norwegian artist, who divides his time between the United States and Norway. In 1996, in viewing a special exhibition of drawings and replicas of Leonardo’s inventions, Mr. Sand became transfixed by the shear beauty and modernity of a bridge the Renaissance master had sketched in a notebook — a bridge he proposed building in Istanbul 500 years earlier. On his return to Oslo, he approached the Norwegian Public Roads Administration offering a partnership in building a variation of Leonardo’s bridge in Norway. Rare among public utility works anywhere, the Norwegian Public Roads Administration is known to operate with a policy of promoting art in public spaces.  Nonetheless, finding receptive officials willing to resurrect a half-millennium old design planned for Turkey and implement it in Scandinavia had to be a pleasant surprise for Mr. Sand. Soon an ideal partnership would develop between the resourceful artist and the enlightened Public Roads Administration.

Gateway Arch, St. Louis, MO (Completed in 1965).

Leonardo’s original design for a single-arch bridge made of stone and long enough to span the 240 m (800-ft) width of the Golden Horn in Istanbul, called for a “pressed bow structure.” This is the mathematical shape described as an “inverted parabola” or “a parabola that spills water.” The parabolic structure supports the weight of the roadway draped over it by pressing down and outward against the bridge’s abutments that are anchored in terra firma. More generally, the silhouette of the bridge in Leonardo’s drawing suggests that a pair of parabolas with different steepness is superposed. The area between the wider parabola, located above, and the steeper parabola (below) creates an elegantly tapered parabola, resembling the upper reaches of the Gateway Arch in St. Louis (built between 1963-65).  The roadway in Leonardo’s drawing is draped over the supporting parabolic structure.


Mr. Sand and the Norwegian Public Roads Administration agreed that Leonardo’s design could be adapted for a pedestrian/bicycle bridge over Highway E-18, connecting Oslo and Bergen and about 20 km southwest of Oslo.  Leonardo’s original specs for the Golden Horn would be scaled down to roughly 1/3 for the Ås site. The roadway would measure 360 feet in length.

Between 1996 and 2001, the Leonardo Bridge Project team performed systematic design and structural load tests in close collaboration with architects and engineers. Both a Leondardesque stone bridge and a laminated-wood version were developed and tested. In the end, the bridge in Ås would be created as a steel-reinforced structure, comprised of laminated wood. It was fortuitous that the technology to build massive structures out of laminated Norwegian Spruce had already been developed in forest-rich Norway. For the 1994 Winter Olympics in Lillehammer, laminated wood had been employed in the construction of a cavernous arena. In the creation of the Leonardo Bridge precisely shaped components could be prefabricated, then assembled with the use of cranes. A set of three parabolic supports were created for the bridge, a central component supporting  the roadway, and two others stabilizing the central parabola. The roadway would rest on metal pillars of varying lengths (visible in the photos below).

A pair of images showing the assembling of the Leonardo Bridge over Highway E-18
A pair of images showing the assembling of the Leonardo Bridge (2001). All the Photos, compliments of the Leonardo Bridge Project.


Night and day view of Leonardo's Bridge over Highway E-18, twenty-km from Oslo.
Night and day views of Leonardo’s Bridge over Highway E-18, twenty-km from Oslo.


After spearheading the building of the Leonardo Bridge in Ås, Mr. Sand and the Leonardo Bridge Team launched a program to build bridges on each of the earth’s continents. Several Leonardo Bridges already have been created of ice.

Ice Bridge in the UN Plaza (2007); Copenhagen (2009); Greenland (2009); Antarctica 2005.
Ices bridges around the world. Top: The UN Plaza (2007); Copenhagen (2009); Bottom: Greenland (2009); Antarctica 2005.

During the first decade of the 21st century, the Leonardo Bridge Project Team built a series of four ice bridges. At any level, they are works of art, far more aesthetic than utilitarian, and far more fleeting than permanent, but they all resonate with strong symbolism. They represent bridging space and time, ethnic and cultural differences, or in Sand’s words, “…touch something eternal.” The four ice bridges seen in the array above were intended to promote the “LIVE ICE” Project, calling attention to the fragility of the Earth’s ice cover. • Ice Bridge in Copenhagen created for the Climate Conference in late 2009. The spire in the background belongs to the Copenhagen Contemporary Art Center (formerly the Church of St. Nikolaj) • Ice Bridge at the UN Plaza, New York, 2007 • Ice Bridge in Greenland in early 2009 • Ice Bridge in Antarctica 2005. An amalgam of water and crushed ice hacked at the site is poured into prefabricated metal forms and allowed to freeze. When the forms are removed, left behind is an ice bridge with the graceful design inspired by Leonardo’s original drawing.


Ultimately, Leonardo’s bridge exhibits the “pressed bow design,” or an “inverted parabola” (in the language of calculus, a “parabola that spills water”). Its shape is described by a simple quadratic equation, including a leading negative sign to insure that the parabola “spills water,” and a coefficient employed as a “Dilation Factor”, that determines the steepness.

“Let no man who is not a mathematician read the elements of my work.” — Leonardo

Leonardo's self-portrait (c. 1510), Turin Royal Library.
Leonardo’s self-portrait (c. 1510), Turin Royal Library.

Leonardo did not know very much formal mathematics, but he had abiding respect and love for mathematics. He would have had no understanding of the analytic geometry and calculus, areas of mathematics that would be formulated during the 17th century by Descartes, Newton and Leibnitz, and used in describing curves, and calculating optimal shapes. Yet he demonstrated an intuitive feel for mathematical shapes and patterns. The hyperbolic spiral appears in the curls of his subjects’ hair, and in the eddies in flowing water. He knew the parabola well, having recognized a century before Galileo that the trajectory of a projectile is a parabola. This is seen in his drawings. He illustrated the book de Divina Proportione (“The Divine Proportion”) for his mathematician friend, Luca Pacioli, demonstrating again that mathematical intuition.

The more one studies Leonardo, the more one realizes that he was in the business, not of predicting the future, but of inventing the future. It would not be until the Industrial Revolution that the materials and the mechanical power would become available for Leonardo’s mental inventions, his dreams, to be fully realized. By the late 19th century the use of  steam and electrical power, of new materials and techniques, of steel girders, high tensile strength cables made his idea more practical. But, Leonardo had realized four hundred years earlier that the parabolic arch would be the key to distributing the weight, especially with the widening footholds. One of the most celebrated bridges of the 20th century, built in the 1930s, is the Sydney Harbor Bridge, spanning a waterway of 503 m (1650 ft). A pair of parabolic structures, one above the other, is used in suspending a flat roadway, rather than supporting it from below. A truss-system of cross-arms in a saw-tooth pattern gives it a more slender overall silhouette, and minimizes the steel required.

Sydney Harbor Bridge, featuring a parabolic system of supports for the suspended roadway. Just to the left, the Sydney Opera House, that appears to echo the bridge with its own parabolic sails. Photo compliments of Murray Lines of Sydney.
Sydney Harbor Bridge, featuring a parabolic system of supports for the suspended roadway. Just to the left, the Sydney Opera House which appears to complement the bridge with its own parabolic sails. Photo, Murray Lines of Sydney.

Daniel Levy had written in a 1999 edition of Time Magazine, “A bridge that Leonardo designed in 1502…is scheduled for construction outside Oslo and there is talk of building similar bridges in Des Moines, Iowa and Istanbul itself.” The bridge in Norway was built, and indeed opened with pomp and circumstance by no other than Queen Sonja in 2001. An array of ice bridges have been built across the world. And Leonardo is one step closer to getting his bridge built in Istanbul. The bridge in Iowa, unhappily, fell through, deemed, “…too modern for our times.”


Melinda Iverson and Vebjørn Sand
Melinda Iverson and Vebjørn Sand

For informative discussions and for photos of the Leonardo Bridge Project I am deeply grateful to my friends Vebjørn Sand and Melinda Iverson.

Vebjørn is the visionary artist who spearheaded the building of Leonardo’s Bridge in Ås, and subsequently the ice bridges in other parts of the world.

Melinda is the International Projects Producer for the Leonardo Bridge Project, Inc., charged with developing the group’s projects around the world and writing, designing and curating events related to the Leonardo Bridge Project/LIVE ICE and Vebjørn’s other public art projects. See the website,

Finally, I am grateful to my friend, Murray Lines, who shared the photo of Sydney Harbor for this article.

So far in this series:

LEONARDO’S BRIDGE: Part 1. “The Master of all Trades”

LEONARDO’S BRIDGE: Part 2. “A Bridge for the Sultan”

LEONARDO’S BRIDGE: Part 3. “Vebjørn Sand and Variations on a Theme by Leonardo”

Still to come: Part 4. “Hakan Kiran and Variations on a theme by Leonardo”

In October 2012 the Turkish Government announced through the public media plans to build Leonardo’s Bridge over the Golden Horn (Haliç). The Istanbul based Architectural Firm of Hakan Kiran, that has been garnering an impressive international reputation for its bold works in Turkey and Europe, was awarded the commission that Leonardo sought in 1502. Hakan Kiran, his wife Tülin Kiran, along with Chief Architect A. Ulvi Altan, have been working on variations of Leonardo’s Bridge. The timing is auspicious with Mr. Altan currently serving as the President of the Mediterranean Society of Engineers and Architects  (SEAMED). The group operates under the auspices of the scientific organization Réseau Méditerranéen des Ecoles d’Ingénieurs (RMEI). Under Mr. Altan’s leadership, SEAMED itself is rapidly growing into a unique institution in its collaborative efforts on the Leonardo da Vinci Bridge Project (LDV Bridge). The group’s motto, “Bridging Cultures and Sharing Heart,” would have pleased Leonardo. The bridge for the Golden Horn is still in design stage, but it is in decidedly good hands!

Bulent Atalay, a scientist, artist and author, has been described by NPR, PBS and the Washington Post as a “Modern Renaissance Man.” He is the author of two successful books on the intersection of art, science and mathematics, with Leonardo, the pre-eminent Renaissance man, serving as the foil. His best selling book, "Math and the Mona Lisa," (Smithsonian Books, 2004) has appeared in 13 languages. Professor Atalay's academic background is in theoretical physics. He travels around the world lecturing at academic institutions and on cruise ships on the "A-subjects," art, archaeology, astrophysics, atomic physics and Ataturk, confessing that he knows much less about the "B-subjects," business, banking, biology and botany... He is the President of the Ataturk Society of America (ASA), dedicated to promoting Ataturk's ideals of science and reason over dogma and superstition, of a secular state with full equality of genders. For more details click on Bulent Atalay
  • Melinda Iverson

    Can’t wait for the next section!! Wonderful!!

  • Caroline Carver

    I loved reading this series about Leonardo’s bridge, his invention of the future, as you so eloquently explain. Norway’s promotion of art in public spaces in partnership with the Public Roads Administration is impressive, and I hope we will see Leonardo’s bridge over the Golden Horn in our lifetime. Remarkable!

    • Thank you for the comment, Caroline. I am impressed by Hakan Kiran and his architectural firm in Istanbul. I think they will do justice to Leonardo’s dream. (See Hakan Kiran Architecture / Bold steps from tradition to the future…

  • John Maenhout

    It surprised me that Leonardo did not know much formal mathematics, and yet he was able to create such designs. Centuries later he inspires others with his designs. Dear Bulent, thanks to your research I have come to a deeper appreciation of the great master Leonardo da Vinci.
    Thank you for your work on National Geographic.

    • Hello John, Leonardo used to tell his apprentices, “Learn from nature, not from each other.” Five hundred years later, we learn from, and appreciate what we can learn from nature. Regarding his mathematics, in the 1490s he put aside painting, in order to learn mathematics from Fra Luca Pacioli. No doubt, he quickly surpassed Pacioli, but it was his intuitive understanding, rather than the formal training that allowed him to bring together the mathematics, art, architecture… His study of anatomy, however, was all his own work. He did anatomical studies (although it was illegal in those days) and illustrated the human body with greater understanding than any anatomist ever. Thanks again, John. Bulent

  • Didi Massoud

    Leonardo da Vinci was a great visionary
    I did not know that this project could be done today
    I appreciate that this architecture is highlighted by the image
    Well done my friend

    • Thank you for the note from the Loire Valley in France. You live close to Amboise, I believe, where this ornament to our species is buried. Unhappily, after his crypt was disturbed during the French Revolution, and his remains were thrown into a common grave, they cannot be identified.

  • Wireless Bridge website

    I just came onto your post and found it quite interesting. I am also associated with IP Camera Software, Ruckus Wireless and enjoy to read the stuff on the same as its rarely found on internet. Thanks again for writing such a good post.

  • Ghislaine Scarici

    Dear Bulent,
    What a fascinating article! Leonardo was a genius. The details you presented are remarkable. Congratulations for your interesting works, you enriched my knowledge.
    Thank you very much.

  • Maria Ocampos

    An informative article from the beginning to the end, with good photos to illustrate it, the one in Oslo is so simple and well illuminated, while the ice ones besides being ecological, are beautiful and ephemeral.

    About Leonardo, what can I said, but remember his words..”Let no man (or women, politically correct) who is not a mathematician read the elements of my work”
    So I only will say that his idea of a bridge made of a single arch over the Golden Horn, was indeed a creative one, lets pray it will be done in harmony with the magic environment
    Thanks for sharing your knowledge about Leonardo!

  • micro job

    I really like and appreciate your weblog post.Much thanks once more. Want more.

  • Jasmine wang

    Hi Bulent,
    How are you?
    A very informative article.. great work.
    I just came across your article, and forward it to a great friend of mine. I took a photo of Mathematical Bridge in July and thought of an article about an old bridge in China I learned at school.
    and one of my friends great uncle built this one :, unforturnately both of them were built without Leonardo’s help, Marco Polo didn’t tell him 😀
    keep up your great work, back soon.


    Valuable post , I was enlightened by the details – Does anyone know where my assistant could get a fillable CMS 1490S example to fill out ?

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