Compass Constructions

Using a compass and straightedge enables precise geometric constructions, allowing for the creation of complex tessellations. To begin these constructions, a single starting point is sufficient. Circles, which represent a collection of points at a specific distance from the center, and straight lines, which represent…

The Koch pentagon is a fractal shape that starts with a regular pentagon. Here, we divide each side of the pentagon into three equal parts. Then, we replace the middle with two sides of an outward-pointing equilateral triangle. We repeat this process for every new…

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…

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…

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…

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 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…

Santiago Calatrava, a celebrated Spanish architect, structural engineer, sculptor, and painter, is renowned worldwide for his groundbreaking and futuristic designs, which frequently fuse architecture with engineering and artistry. Among his many notable works is the door design at Ernsting Warehouse in Coesfeld, Germany, constructed in…

While digging through the lecture archive, I found this video I made in 2017. We introduce Platonic solids and Archimedean solids in the Design Geometry course at Istanbul Bilgi University. This video shows how we can create an Archimedean solid, the Truncated Tetrahedron, by folding…

This is a new paper published in Nexus Network Journal. I tried to implement euclidean constructions by compass in the approximation of famous parametric curves; Bézier curves, and B-Splines. Then, I created the algorithms to calculate the number of steps on a compass-only construction of…

This video shows the method of constructing equal segments on a line by only using compass operations. Here, you can see that there is a pattern in the number of required compass operations. For example, if you need to divide the distance by 3, you…

I learned this method from the open math resources website. I couldn’t help myself repeat it in Rhinoceros. It was quite fun to solve circle tangency problems in 2D, this is one of them: drawing the circle that passes three given points, not using ready-made…

Here is a simple explanation of the famous Euclidean Constructions: Why didn’t Euclid just measure things with a ruler and calculate lengths? For example, one of the basic constructions is bisecting a line (dividing it into two equal parts). Why not just measure it with…

The first-year Architectural Geometry course includes Euclidean constructions as a study of associative geometry. We have exercised the below questions to study this topic. These are three mutually tangent circles, that can be drawn using only a compass and ruler, without built-in tangency functions in…