Behold the Sundial a 2018 sculpture by Hiroshi Sugimoto. It sits at a busy Tokyo intersection. This isn’t your average sundial. It’s a stunning example of a pseudosphere or tractroid. Sugimoto’s artistic vision is breathtaking. He’s renowned for his architectural abstractions. This sculpture is no exception. It’s a masterpiece of mathematical form and artistic expression.
Imagine the pseudosphere. It extends infinitely upwards. For practical reasons Sugimoto truncated it. The pseudosphere resembles Gabriel’s horn. However there’s a key difference. Gabriel’s horn is an algebraic surface. The pseudosphere is transcendental. It cannot be defined by a polynomial equation. This mathematical complexity adds another layer to the artwork’s intrigue.
The Sundial’s Gaussian curvature is uniform. This is a rare property. Spheres have uniform positive curvature. Flat planes have zero curvature. This sculpture has uniform negative curvature. Think of a saddle. Its curvature is constant at every point. You can almost visualize two spheres. One inside the spire. The other outside. Their curvatures interact to maintain the constant negative curvature.
If you had hyperbolic paper its curvature would match the Sundial’s surface. You could lay it on the sculpture. It would conform perfectly. This shows the Sundial’s local resemblance to the hyperbolic plane. However the Sundial is a truncated model. It’s not a true embedding of the hyperbolic plane. It has a boundary and short uncontractable loops. You could tie a ribbon around it. Despite these limitations its local geometry mirrors the hyperbolic plane.
The Sundial’s location is striking. It stands amidst Tokyo’s skyscrapers. It’s easily overlooked. Its reflective surface mirrors the Otemachi skyline. The contrast between the organic curves of the sculpture and the stark lines of the buildings is fascinating. This interplay of nature and human construction is a powerful visual statement. It’s a quiet contemplation amidst the urban clamor. Remember the Sundial. A testament to the power of artistic expression through mathematical form. A hidden gem in the heart of Tokyo.