Studios

Curvature can be roughly described as how much a curve is “turning” at point a P. We place two “very” close tangents and measure the difference between them. The closer these tangents are, the more precise our approximation would be. An osculating circle is a tangent circle that has the same curvature as the curve at point P. The larger the circle, the more “flat” the curve is. An infinitely large osculating circle means the curve is linear at point…

Truncation refers to the process of shortening something by removing parts. You can apply truncation to numbers, text, or data in various contexts. A truncated polyhedron is a geometric solid formed by truncating the vertices of a regular polyhedron. Truncation involves cutting off the corners or vertices of the polyhedron in such a way that the original faces become polygons with new edges. This process creates new faces at the truncated vertices. It often results in a shape that is…

Here is a method for coding the dodecahedron and all its irregular variants in Grasshopper as quickly as possible. I utilized the golden ratio rectangles, usually used to construct the sister polyhedron, the icosahedron. However, the magic component of the Grasshopper, the Faceted Dome rescued me again to generate the dual of it, the dodecahedron. This is a special platonic solid, which has 12 regular pentagonal faces. There are several techniques that I generally prefer to use in Grasshopper. You…

The rhombitrihexagonal tiling is one of the semi-regular tessellations. It is composed of regular hexagons, squares, and triangles. It is a periodic tessellation since you can copy the fundamental unit and move it across the plane to generate the tiling. I use this quality of the tiling to draw and expand it in Rhinoceros software. This is a basic drawing exercise. At the same time, it is a nice exercise for the mind and sometimes for the soul. There are…

The snub square tiling is one of the semi-regular tessellations, where regular triangles and squares match perfectly to fill the plane without gaps or overlaps. The Euclidean construction of Snub Square tiling is possible by utilizing the basic compass and straightedge operations. I made this construction in Rhinoceros to show that there is no need for any numerical input to locate the points and draw the tiling. There are two major techniques explained there. One of them is the basic…

Villa Savoye is an iconic modernist villa in Poissy, France, built between 1928 and 1931. It was designed by the pioneering Swiss-French architect Le Corbusier, often considered one of the most influential architects of the 20th century. Villa Savoye is a masterpiece of the International Style, embodying the principles of modern architecture that prioritize functionality, simplicity, and the use of new materials such as reinforced concrete. I made this digital model for a research project completed at YTU in 2007….

This is a 3d modeling tutorial for the platonic solid of dodecahedron. Modeling a dodecahedron is a good exercise for the basic transformation commands such as Rotate3D in Rhinoceros. You will see that it is possible to calculate the rotation angle by using sphere intersections. I learned this elegant method while teaching Architectural Geometry classes 12 years ago. It is based on the fact that, given a rotation axis and the rotation point, you can draw the path traced by…

This is a short video tutorial on the B-Spline decomposition I studied earlier here. This tutorial demonstrates how to decompose a B-Spline curve into Bezier curves using Rhino. Despite the original Bezier-de Casteljau algorithm requiring degree+1 control points, Rhino allows drawing a degree-3 curve with any number of control points. By examining knot points and dividing segments appropriately, the B-Spline curve can be manually subdivided into Bezier curves. This involves placing control points, dividing segments, and connecting points in a…

In this short tutorial, I am going to show you how to locate a parametric point on a Bezier curve. This will be a third-degree cubic Bezier curve. So, I start by placing four control points. I name these points from P0 to P3. Then, I connect them by a polyline in order. I explode the polyline into the segments. The parameter of my point must be a number between 0 and 1. I will divide the segments, so I…

“Mihr-î Mah” in Ottoman Turkish translates to “The Sun and The Moon” in English. She was the most powerful imperial princess in Ottoman history according to historians. The Mihrimah Sultan Mosque is located in Üsküdar district of Istanbul. It was built for Mihrimah Sultan, the daughter of Kanuni Sultan Süleyman and Hürrem Sultan. It is one of the early works of the architect Mimar Sinan. Sinan supported the main dome with half domes on three sides, excluding the front facade….