In geometry, “tangent” refers to a scenario where a line, curve, or surface touches another at exactly one point without crossing or leaving a gap. A tangent line to a circle is a straight line that touches the circle at only one point, and it is perpendicular to the radius at that point. Two circles can be tangent either internally or externally. Geometric constructions using a compass and straightedge can help accurately calculate tangent circles. Thus, the “constructing tangent circles”…

The geometric shapes constructed using only a compass and straightedge have significantly influenced the development of reasoning and logic that underpin modern science. This non-numerical approach is also known as the Synthetic Geometry. Before René Descartes introduced Analytic Geometry, this method was widely studied and practiced. In “Elements”, Euclid explores fundamental rules, such as ‘The straight line between two points is the shortest,’ and derives all geometric truths from simple, observable principles, built up step by step through logical deduction….

The practical geometry of the ancient Egyptian “rope-stretchers” delineated land after the annual flooding of the Nile River. Thus, the term “geometry” derives from the Greek words “gaia” (earth) and “metria” (measurement). Greek mathematicians used compass and straightedge to perform similar calculations on paper. This abstract thinking allowed for insights into the underlying logic, independent of the accuracy of hand tools. So, the abstract “circle” and “line” constructed by perfect compasses and straightedges can be seen as a reflection of…

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…

Below is the first paper of my son, Mete Yazar. It is about a mathematical and geometric exercise of calculating the surface area of an arbitrary shape (a classical guitar’s body panel). He did a good job in utilizing Bezier/de Casteljau curves and generating the parametric equations of the piecewise curve. I helped him to validate the results by using rhinoceros CAD software. Therefore, it seems that his calculations are correct. In this paper, we attempted to calculate a guitar’s…

ASCII art is a graphic design technique that uses characters from the ASCII (American Standard Code for Information Interchange) set to create images, symbols, and designs. This form of art involves arranging text characters to form a visual representation of objects, scenes, or abstract patterns. I first encountered this art form in the 90s through readme text files and computer games. Years later, attempting to automate it in Grasshopper was a lot of fun. My initial goal was to calculate…