Studios

A spiral is a curve that begins at a central point and continuously moves outward, either getting larger (in an outward spiral) or smaller (in an inward spiral). It often revolves around a central point or axis. Spirals are fascinating because they appear in many aspects of nature, mathematics, and art. In this drawing exercise, I utilized several commands to draw such a system. However, this drawing is not a curved spiral but rather a segmented one. In this basic…

A fractal is a complex geometric shape that we can split into parts, each of which is a reduced-scale copy of the whole. This property is self-similarity. Fractals often exhibit patterns that repeat at different scales. We can find fractals in nature, such as in snowflakes, mountain ranges, trees, and clouds. We can also generate them mathematically. Fractals have applications in various fields such as physics, biology, and computer graphics. We use them to model natural phenomena, compress data, and…

In mathematics, “curve” describes one-dimensional objects or line shapes, regardless of their curvature. Straight lines, polylines, and curved lines all fall under the category of “curves.” You may remember working with equation graphs in high school math classes. For instance, a first-degree equation produces straight-line graphs, while higher-degree equations, like “x squared,” create curved graphs. In this context, we focus on degree-1 curves, drawing straight-line segments using the polyline command. In Rhinoceros, the software refers to all objects, whether curved…

The Flowsnake, or Gosper curve is a space-filling fractal. It is also known as the Peano-Gosper curve. There are other similar space-filling fractals such as the Dragon curve, or the Hilbert curve. A space-filling fractal is a special type of curve, that fills a plane when iterated infinitely. This means, that if you continue to replace every segment of the polyline with the whole drawing, it will quickly become very dense and ultimately fill the plane without any gaps or…

In computer-aided design (CAD), a polyline is a series (or a chain) of straight lines. Each straight section of a polyline is a “segment,” and the points where the segments connect are “vertices.” If a polyline’s starting and ending vertices coincide, it is a “closed polyline” or a “polygon”. If they do not, the polyline is classified as an “open polyline.” Polylines can be planar (2D) or 3D. The Gosper curve, named after Bill Gosper, also known as the Peano-Gosper…

Modeling a rhombicosidodecahedron requires exploding and extending the faces of a dodecahedron and an icosahedron of the same edge length. We begin with both polyhedra centered at the same point. Then, we explode the faces of the dodecahedron and icosahedron outward from the center. We extend their planes while maintaining their orientation and shape. As these faces extend, they intersect and form new polygonal regions. Triangular and pentagonal faces emerge naturally from the extended faces of the dodecahedron and icosahedron,…

Gyroid is a popular triply-periodic minimal surface. Although it is a mathematical entity, designers and architects like its approximations very much. We used an interpretation of Gyroid in the rammed-earth structure: “Common-action Wall” in 2017. In that project, we utilized its spatial quality of dividing the space into two intertwining and symmetrical volumes. By making one of these volumes solid, I left the other void. So, in this tutorial, I am going to explain modeling the Gyroid Solid by utilizing…

In this short tutorial, I am showing the essentials of data domains in Grasshopper. A domain is a data type in Grasshopper that represents a portion of the 1d or 2d number space. This requires a starting and ending point in those spaces. In 1d, these points are two numbers. Then, the domain represents all the numbers between them. In 2d, a data domain is represented by two coordinates (u, v). This presentation illustrates this concept with the area defined…

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…