Although we already have a connection on hyperboloid structures, there is still none that connects the sites with specific mathematical concepts. Architecture in itself is highly mathematical discipline. However, this connection will relate the sites with a specific mathematical concept - either the site is a good example of the mathematical concept or it demonstrates development of complex mathematical concepts. We can exclude sites with ingenious use of proportions, the golden ratio and symmetry (unless they exhibit other higher mathematical concept) as they basic concepts in architecture. are excluded unless connected to a higher mathematical concept.

For Golden ratio, we can have a separate connection as use of it on some sites are still debatable.

**Hampi: Virupaksha Temple** -

*Fractals*. Fractals are complex patterns that shows repetition across different scales. Virupaksha temple exhibits fractals in temple carvings and patterns as well as in its layout. "As you look up the temple top, the patterns divide and repeat themselves, just like you would see in a snowflake or some other natural wonders."

http://www.blankslatechronicles.com/virupaksha-temple-hampi-india-story-mathematical- wonder/**Granada: Alhambra** -

*Tessellation*. Tessellation is the periodic tiling of a plane with no overlaps or gaps. The Alhambra is the classic example of uses of tessellation in architecture. In particular, the Alhambra tiles contain "nearly all, if not all, of the 17 mathematically possible wallpaper groups."

http://www.math.nus.edu.sg/aslaksen/teaching/math-art-arch.shtml#Symmetry,

https://en.wikipedia.org/wiki/Alhambra, [url=https://en.wikipedia.org/wiki/Wallpaper_group ][/url]

**Sydney Opera House** -

*Spherical Geometry*. The shells are derived from triangular sections of a surface of sphere with similar radius. Known as the "Spherical Solution" it became "the binding discovery that allowed for the unified and distinctive characteristics of the Sydney Opera House to be realised"

https://www.sydneyoperahouse.com/our-story/sydney-opera-house-history/spherical-solut ion.html**Works of Antoni Gaudí: Sagrada Familia and Parc Guell **- Conic Sections and Ruled Surfaces. Conic sections are curves obtained when a plane intersects a right circular cone forming circle, ellipse, parabola and hyperbola. Sagrada Famila and Parc Guell have several elements of parabolic and hyperbolic nature.

Ruled surfaces are surfaces that are formed by moving a line in space. Some of the windows at Sagrada Famila exhibits Hyperboloids of One Sheet.

http://mathstat.slu.edu/escher/index.php/The_Geometry_of_Antoni_Gaudi**Central University City Campus of UNAM: Pabellón de Rayos Cósmicos** -

*Hyperbolic Paraboloid*. Hyperbolic paraboloid is a doubly ruled surface shaped like a saddle. "Cosmic Ray Pavilion is the product of the structural experiments of the architect Felix Candela is a very thin reinforced concrete double curvature based on the geometry of the hyperbolic paraboloid."

https://en.wikiarquitectura.com/building/Cosmic-Ray-Pavilion/**Brasilia: Cathedral of Brasilia** -

*Hyperboloid of revolution*. Hyperbolic of revolution are surfaces generated by rotating a hyperbola around an axis. The shape of the roof of the cathedral is hyperboloid of revolution with asymmetric sections and consists of 16 identical columns.

https://en.wikipedia.org/wiki/Cathedral_of_Bras%C3%ADlia**Seokguram Grotto and Bulguksa Temple: Seokguram Grotto** -

*Projective Geometry*. Projective geometry is the study of mathematics dealing with properties of figures that is invariant (doesn't change) with projective transformations. Seokguram Grotto demonstrates the practical knowledge in projective geometry of the builders. Viewed from around 10 meters (current location with the glass door), the grotto is eye-level with the visitor and with balanced in principle with projective geometry. The halo is constructed as an ellipse but after geometric projection around 8-10 m, it looks round from the observer. The dimensions of the two hands are constructed differently due to the distance difference between them. When viewed from observer, the hands look perfectly balanced. Same principles is also applied to the dimension of the head in relation to the body.

https://www.coursera.org/lecture/silla-science-technology/visitors-overview-qZtiu**Divrigi** -

*Projective Geometry*. At different hours of a day, 4 silhouettes appear on exterior walls (a man looking straight, man reading a book, a man praying, a woman praying). "These remarkable features could not have been designed without the combination of mathematics, astronomy and art... before the construction of the mosque had started, the scientists observed the positions of the sun and stars for two years. After very careful calculations had been done, the results were applied in the construction of the walls and the carving of the outside doors."

http://www.muslimheritage.com/article/new-discoveries-in-islamic-complex**Gonbad-e Qābus** -

*Cycles and Circumferences*. "The monument is an outstanding example of an Islamic commemorative tower whose innovative structural design illustrates the exceptional development of mathematics and science in the Muslim world at the turn of the first millennium AD." - statement of OUV

http://edizionicafoscari.unive.it/media/pdf/books/978-88-6969-101-0/978-88-6969-101-0 -ch-05.pdf**Castel Del Monte** -

*Fibonacci Sequence*. "The floors, fireplaces, windows, everything seems to follow the sequence of the famous mathematician Fibonacci."

http://www.apuliavillas.com/castel-del-monte-puglia-between-fibonacci-esotericism-and -history-2/