I learnt 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 commands but only geometric tools of circle (compass) and ruler (line). Here is the sequence of it: First of all, we need to know that the circle we are looking for is centered at somewhere on the perpendicular paths between the points. This means, we […]

## euclidean construction

Here is a simple explanation about 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 a ruler and divide by two? One theory is the the Greeks could not easily do arithmetic. They had only whole numbers, no zero, and no negative numbers. This meant they could not for example divide 5 by 2 and get 2.5, because 2.5 is […]

First year Architectural Geometry course includes euclidean constructions as a study of associative geometry. We have exercised below questions to study this topic. These are three mutually tangent circles, that can be drawn using only compass and ruler, without built-in tangency functions in Rhino. Such exercises are expected to improve students’ reasoning. We believe architectural geometry education should encourage a conception that allows students to think about what they are doing and control their process, rather than just using commands to produce forms randomly.

Truncated Icosahedron (5,6,6) is an Archimedian Solid, probably the most popular one because of it’s apperance as the soccer ball. It’s constructed by trimming one third of each edge of an Icosahedron, (a Platonic Solid described here). In order to find 1/3’s of each edge, I used duplicate border, explode and divide commands to get the points that construct the pentagons and hexagons, while paneling is done by using planar surface commands on closed polyline edges. (If you are interested in how we reached the initial solid, refer to the […]

Octahedron is a platonic solid with 8 faces of identical equilateral triangles. It has a close relationship with cube as it’s dual. In order to construct an octahedron, we first have to create a square. Main problem of drawing the square is determining the right angle (perpendicular axis) to any point in euclidean space. We’ll draw it here as a two dimensional projection. However it can also be established in three dimensions with the same method (except using spheres instead of circles). Start from any two point in the space; […]

Dodecahedron is a Platonic Solid with 12 equilateral pentagonal faces. It has a close relationship with it’s 20-sided dual, Icosahedron. Mete Tüneri showed the following method of Dodecahedron construction, using only distances, corners of pentagon and a visionay equilateral triangle underneath. We’ll construct Dodecahedron, assuming that we’ve drawn an initial equilateral pentagon. We need to find out the pentagon’s angle of 3d rotation. First, put spheres at points a and c, with the radius of a to c. Intersection of these spheres result a circle. We know that, every point on […]

Icosidodecahedron is an Archimedian Solid, a thing in between the Platonic Solids of Icosahedron (d20) and Dodecahedron (d12). It is a rectified version of Icosahedron, constructed with dividing every edge of it into two equal segments, and joining these segments to create a composition of equilateral pentagons and triangles. Archimedian Solids consists of at least two equilateral polygons, whereas Platonic Solids are constructed by only one. We’ll deduce an Icosidodecahedron from Icosahedron below; First, you should create an Icosahedron, the Platonic father of Icosidodecaheron. After that, all faces should be […]

Icosahedron is one of the five Platonic Solids with twenty equilateral triangular faces. It’s dual is Dodecahedron, which has pentagonal faces. Here, Icosahedron is constructed by using pentagons. Interesting thing is it’s close relationship with Dodecahedron, although they seem to be very different. This time we won’t lose time with two dimensional pentagon drawing. Maybe we’ll discuss that later. Assuming you’ve created a regular pentagon, you should find the “tip” point of the Icosahedron by intersecting spheres from at least three of the corner points with a radius of pentagon’s […]

Truncated Tetrahedron is an Archimedian Solid, created by slicing a Tetrahedron. It’s faces are regular hexagons and triangles. Assuming you’ve created a Tetrahedron, first join it’s faces to create a polysurface. Now, you may re-create the lines of Tetrahedron’s edges, either by drawing them or generating them (Curve/Curve from Objects/Duplicate Edge). While the edge lines are selected, hit (Curve/Point Object/Divide Curve By/Number of Segments) and type 3 to the number of segments to be created. Now all edges should be divided equally into three parts. Draw the equilateral triangles, connecting the […]

Tetrahedron is a platonic solid with 4 equal triangular faces (which are also equilateral), 6 equal edges and 4 vertices. While creating this shape, we will take a closer look at length transfers using compass-like tools both in two and three dimensional space. In order to define the edge length of first triangle (which is a straight line), start with any two points in cartesian space. Using compass (arc or circle), draw two arches (or circles) using your initial points as corners, and the distance between your points as radius. […]