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…

Geodesic refers to the shortest path between two points on a curved surface. It is based on the principles of geodesy, which is the science of measuring the Earth’s shape. On the other hand, in architecture and design, a geodesic dome is a spherical or hemispherical structure consisting of a network of geodesic lines (great-circle arcs) forming triangles. Therefore, the dome’s framework provides strength and stability, distributing stress throughout its structure. Buckminster Fuller popularized and developed the concept of the…

Stellated polyhedra are three-dimensional geometric shapes formed by extending the faces of a regular polyhedron (a solid with flat faces) beyond their original boundaries until they intersect with each other. The term “stellate” comes from the Latin word “stella,” meaning star and these polyhedra often have a star-like appearance due to their extended faces. They are popular because of their aesthetic qualities. I studied these forms many times before. This time, I implemented the stellations and negative stellations (the excavated…

I have been studying the Bezier-De Casteljau algorithm for several years. Last week, I finally managed to understand and implement B-Spline Construction in Rhino and Grasshopper Python code, using the Cox-De Boor algorithm. This algorithm calculates the effect of control points on a B-Spline with many control points. After this, I am not limited to the degree+1 control points. This animation shows both algorithms working together. This Rhino Python code in a Grasshopper definition can handle three tasks. The first…

In 2016, archi-union architects and fab-union intelligent engineering completed the renovation of the art gallery in Shanghai, China. The distinctive feature of the building was the robotic masonry fabrication of the brick facades. The undulating and waving parametric bricks were increasingly becoming popular after the introduction of parametric design tools such as Grasshopper and the works of Gramazio & Kohler at ETH Zurich since 2008, I guess. I made two Grasshopper experiments in 2012 and 2013 to generate such structures….

“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….

The Hilbert Curve, also referred to as the Hilbert space-filling curve, was initially introduced by the German mathematician David Hilbert in 1891. It is a continuous fractal curve, presenting a variation of the space-filling Peano curves uncovered by Giuseppe Peano in 1890. After a study on the mathematical background of this curve, I implemented a Python code into Grasshopper Python. However, I wanted to explore more variations by playing with the algorithm. This is why, the resulting Grasshopper definition can…

Catalan Solids are the duals of Archimedean Solids. They were first described by mathematician Eugène Charles Catalan in the 19th century. There are 13 Catalan solids, and they exhibit interesting symmetries and unique characteristics. While coding the vertex coordinates of these solids in Grasshopper, I made a simple lamp design to exercise the programming language. The code generates the 13 Catalan Lamps with flaps. Since the polyhedra have planar faces, they can be cut out by hand or a laser…

While reviewing past Grasshopper studies, I stumbled upon a Parametric Muqarnas study dating back 11 years. Although it shares numerous characteristics with a muqarnas design, it is not a perfect match. The Grasshopper definition used in that study was not particularly efficient, prompting me to revisit and improve it. This particular design presents various challenges, making it an excellent exercise for mastering Grasshopper. My goal with this study was to create Parametric Muqarnas Blocks suitable for 3D printing. To achieve…

The interlocking frame structures represent a quintessential example of parametric design since very old times. More recently, they gained widespread attention through the Metropol Parasol in Seville, captivating architects and designers. I’ve extensively explored these structures on this blog, attempting various methods like manual slit drawing and solid/region boolean operations. My latest iteration aimed for greater simplicity and adaptability on complex surfaces. I call this new version Interlocking Pavilion. The ribs now align perpendicularly to the surface, which is a…