A Euclidean Construction of the Golden Rectangle

Start with points A and B. Construct a circle, centered on A, with radius AB. Draw line AB, which intersects this circle at B and C. Construct a second line which is perpendicular to the first line, intersecting it at A. Let the two intersections of the circle and this second be named points D and E.

Bisect segment AB, and call its midpoint F. Construct a line containing D and F. If the circle’s radius is two, then AF = FB = 1, while AD = 2. By the Pythagorean Theorem applied to right triangle DAF, then, DF = sqrt(5). Construct a second circle centered on F, with radius DF. Construct point G where this circle intersects segment AC. It follows that FG, being another radius of this second circle, has a length of sqrt(5). This makes BG = FB + FG = 1 + sqrt(5).

Construct the line which is perpendicular to line AE and passes through point E. Next, construct the two lines perpendicular to line AB and passing through points G and B. These two lines intersect the first line described in this paragraph at two points: H (below G) and I (below B). ABIE is a square with edge length two, and both GBIH and AEHG are golden rectangles.

A Faceted Icosidodecahedron with 48 Faces

Faceted Icosidodeca golden rectangles dodecagons and star pentagons

The 48 faces of this faceted icosidodecahedron are:

  • 6 green regular decagons (difficult to see, because they pass through the figure’s center)
  • 30 golden rectangles, shown in yellow, and each interpenetrating other faces
  • 12 blue star pentagons

Software called Stella 4d was used to make this, and it is available (with a free trial download available) at this website.

A Euclidean Construction of a Golden Rectangle in Which All Circles Used Have Radius One or Two

There is more than one way to construct a golden rectangle using the Euclidean rules, but all the ones I have seen before use circles with irrational radii. This construction, which I believe to be new, does not use that shortcut, which helps explain its length. The cost of avoiding circles of irrational radius is decreased efficiency, as measured by the number of steps required for the entire construction.

In the diagram below, the distance between points A and B is set at one. All of the green circles have this radius, while the magenta circles have a radius exactly twice as long.


To make following the construction from the diagram above easier, I named the points in alphabetical order, as they appear, as the construction proceeds. The yellow rectangle is the resulting golden rectangle. The blue right triangle is what I used to get a segment with a length equal to the square root of five, which is a necessary step, given that this irrational number is part of the numerical definition of the exact value of the golden ratio (one-half of the sum of one and the square root of five). In order to make the hypotenuse have a length equal to the square root of five, by the Pythagorean Theorem, the two legs of this triangle have lengths of one and two.

A Virtual Zomeball


For physical modeling of polyhedra, I often use Zometools (available at http://www.zometool.com), which use Zomeballs as nodes for a ball-and-stick modeling system. To make virtual models such as the one above, though, I use Stella 4d: Polyhedron Navigator (available at http://www.software3d.com/Stella.php).

It occurred to me to try to make a virtual model of a Zomeball, which is one of two equally-symmetrical versions of a rhombicosidodecahedron, with its squares replaced by golden rectangles. If you visit the Zometools page, you can see the way they picture Zomeballs, and then let me know how good a simulation I created, above.

Thirty Golden Rectangles, Rotating About a Common Axis

The third image in the last post is a faceting of the icosidodecahedron. In that faceting, the faces used are equilateral triangles, star pentagons, and golden rectangles. To make these two new images, starting with that particular faceting of the icosidodecahedron, I rendered its triangles and star pentagons invisible, leaving only the thirty golden rectangles. It’s shown twice below, simply because I wanted to show it using two different coloring-schemes.



I would not be able to create images like this without the use of my favorite computer program, Stella 4d, written by a friend of mine who lives in Australia. You can try this program yourself, as a free trial download, at http://www.software3d.com/Stella.php.

Polygons Related to the Golden Ratio, and Associated Figures in Geometry, Part 2: Quadrilaterals

The golden ratio, also known as φ, has a value of [1 + sqrt(5)]/2, or ~1.61803. It is associated with a great many figures in geometry, and also appears in numerous other contexts. The most well-known relationship between a geometric figure and the golden ratio is the golden rectangle, which has a length:width ratio equal to the golden ratio. An interesting property of the golden rectangle is that, if a square is removed from it, the remaining portion is simply a smaller golden rectangle — and this process can be continued without limit.

golden rectangle

While the golden ratio is related to many polyhedra, this relationship does not always involve golden rectangles, but sometimes it does. For example, it is possible to modify a rhombicosidodecahedron, by replacing that figure’s squares with golden rectangles (with the longest side adjacent to the triangles, not the pentagons), to obtain a “Zomeball” — the node which is at the heart of the Zometool ball-and-stick modeling system for polyhedra, and other phenomena. The entire Zome system is based on the golden ratio. Zome kits are available for purchase at http://www.zometool.com, and this image of a Zomeball was found at http://www.graphics.rwth-aachen.de/media/resource_images/zomeball.png.


In some cases, the relationship between a golden rectangle, and a polyhedron, is more subtle. For example, consider three mutually-perpendicular golden rectangles, each with the same center:


While this is not, itself, a polyhedron, it is possible to create a polyhedron from it, by creating its convex hull. A convex hull is simply the smallest convex polyhedron which can contain a given figure in space. For the three golden rectangles above, the convex hull is the icosahedron, one of the Platonic solids:


In addition to the golden rectangle, there are also other quadrilaterals related to the golden ratio. For example, a figure known as a golden rhombus is formed by simply connecting the midpoints of the sides of a golden rectangle. The resulting rhombus has diagonals which are in the golden ratio.

golden rhombus

One of the Archimedean solids, the icosidodecahedron, has a dual called the rhombic triacontahedron. The rhombic triacontahedron has thirty faces, and all of them are golden rhombi.

Rhombic Triaconta

There are also other polyhedra which have golden rhombi for faces. One of them, called the rhombic hexacontahedron (or “hexecontahedron,” in some sources), is actually the 26th stellation of the rhombic triacontahedron, itself. It has sixty faces, all of which are golden rhombi.

Rhombic Triaconta 26th stellation

Other quadrilaterals related to the golden ratio can be formed by reflecting the golden triangle and golden gmonon (described in the post right before this one) across each of their bases, to form two other types of rhombus.

rhombi for penrose tilings

In these two rhombi, the golden ratio shows up as the side-to-short-diagonal ratio (in the case of the 36-144-36-144 rhombus), and the long-diagonal-to-side ratio (in the case of the 72-108-72-108 rhombus). These two rhombi have a special property:  together, they can tile a plane in a pattern which never repeats itself, but, despite this, can be continued indefinitely. This “aperiodic tiling” was discovered by Roger Penrose, a physicist and mathematician. The image below, showing part of such an aperiodic tiling, was found at https://en.wikipedia.org/wiki/Penrose_tiling.


There are also at least two other quadrilaterals related to the golden ratio, and they are also formed from the golden triangle and the golden gnomon. The procedure for making these figures, which could be called the “golden kite” and the “golden dart,” is similar to the one for making the rhombi for the Penrose tiling above, but has one difference: the two triangles are each reflected over a leg, rather than a base.

kite and dart for for penrose tilings

In the case of this kite and dart, it is the longer and shorter edges, in each case, which are in the golden ratio — just as is the case with the golden rectangle. Another discovery of Roger Penrose is that these two figures, also, can be used to form aperiodic tilings of the plane, as seen in this image from http://www.math.uni-bielefeld.de/~gaehler/tilings/kitedart.html.


There is yet another quadrilateral which has strong connections to the golden ratio. I call it the golden trapezoid, and this shows how it can be made from a golden rectangle, and how it can be broken down into golden triangles and golden gnomons. However, I have not yet found an interesting polyhedron, not tiling pattern, based on golden trapezoids — but I have not finished my search, either.

golden trapezoid

[Image credits:  see above for the sources of the pictures of the two Penrose tilings, as well as the Zomeball, shown in this post. Other “flat,” nonmoving pictures I created myself, using Geometer’s Sketchpad and MS-Paint. The rotating images, however, were created using a program called Stella 4d, which is available at http://www.software3d.com/Stella.php.]