Polygons

"Without Geometry life is pointless."
In "King of Infinite Space: Donald Coxeter, the Man Who Saved Geometry", by Siobhan Roberts.

In this site, I will be showing many geometrical models made using the Zometool system. For this reason, I will explain some of its features that will be useful for understanding what follows. The Zometool models in this particular page were made for illustrative purposes only, for that reason they have been disassembled.

Points, line segments, Zometool

 In Fig. 2.1, I show the standard Zometool components, these represent zero and one-dimensional elements, points and line segments.

Geometry starts with geometric points, of zero dimensions. These are represented in Zometool by the connectors, or balls . All balls are identical: they have 12 Pentagonal holes, 20 Triangular holes and 30 rectangular holes, in a pattern that is similar, but not identical, to the faces of the Rhombicosidodecahedron. They have, therefore, an overall Icosahedral symmetry. They are normally provided in white, as in this Figure, but Zometool sells them in a variety of colours.


Fig. 2.1: The standard Zometool components. The balls are in white, the struts are shown for sizes 0, 1 and 2 for the different colours (see explanation below for these).


In geometry, the one-dimensional objects include line segments, which in Zometool are represented by the struts. These are always identical at both ends: We will keep this B, R, Y, G terminology to indicate the shapes of the struts, even when referring to struts in non-native colours, which are also sold by Zometool. The R and Y struts have a twist in the middle, which means that the balls at both ends have always identical orientations. This means that, in any Zometool model, all balls are aligned exactly with each other!

All of these struts come in three regular sizes (n = 0, 1, 2), all shown in Fig. 2.1, with the distance between the centres of balls attached at both ends (which we will designate as its length) of size n being φ times larger than for size n − 1 (we will calculate this number φ soon). Red struts also have a very short (00) size. Green struts also come in half-sizes (HG), two HG2 struts are shown right at the top of the Figure. Size 3 struts were made by Zometool in the early days, but they have been discontinued, and for that reason they are a bit harder to find. I buy them mostly on eBay.

To calculate the lengths of these struts, we will now define the length of the B1 strut as the unit of measurement. Given the geometric progression mentioned above, the B0 strut has a size 1/φ and the B2 strut has a size φ.

Fig. 2.1 shows a very important feature of the system: size 2 struts are as long as the sum of the lengths of struts with sizes 0 and 1. Replacing the values for their lengths, we get an equation (which is strictly for the B struts, but is valid for the other cases as well):

φ = 1 + 1/φ   (a)

Multiplying all terms by φ and moving them to the left, we obtain a simple quadratic equation:

φ2 − φ − 1 = 0   (b)

Using the quadratic formula, we see that there is a single positive solution to this equation:

φ = (√ 5 + 1) / 2 = 1.618 033 988 749 894 848 204 586 834 365...

This is the famous Golden ratio. This is a fundamental number that appears extremely often in Zometool models and in geometry in general. This number has an astounding number of properties, and for this reason we will discuss it in more detail now. From equations (a) and (b), we derive:

1 / φ = φ − 1 = 0.618 033 988 749 894 848 204 586 834 365...   (c)

φ2 = φ + 1 = 2.618 033 988 749 894 848 204 586 834 365....   (d)

These are the lengths (relative to the B1 strut) of respectively the B0 and B3 struts. Using eq. (d), we can calculate the following powers of φ (theoretical Zometool sizes 4, 5, 6, 7):

φ3 = 2 φ + 1 = 4.236 067 977 499 789 696 409 173 668 731...

φ4 = 3 φ + 2 = 6.854 101 966 249 684 544 613 760 503 096...

φ5 = 5 φ + 3 = 11.090 169 943 749 474 241 022 934 171 828...

φ6 = 8 φ + 5 = 17.944 271 909 999 158 785 636 694 674 925...,

which merely extends what Fig. 2.1 is showing, that each term in the series is the sum of the two previous ones. This also happens necessarily to the integer coefficients in boldface, they are the famous Fibonacci numbers. Note that the unit coefficients are one position behind in the sequence relative to the φ coefficients, let's call them respectively F(n − 1) and F(n). Thus, in general:

φn = F(n) φ + F(n − 1).   (e)

Using eqs. (c) and (d), we can calculate the negative powers of φ (theoretical Zometool sizes −1, −2, −3, −4, −5):

φ−2 = − 1 φ + 2 = 0.381 966 011 250 105 151 795 413 165 634...

φ−3 = + 2 φ − 3 = 0.236 067 977 499 789 696 409 173 668 731...

φ−4 = − 3 φ + 5 = 0.145 898 033 750 315 455 386 239 496 903...

φ−5 = + 5 φ − 8 = 0.090 169 943 749 474 241 022 934 171 828...

φ−6 = − 8 φ + 13 = 0.055 728 090 000 841 214 363 305 325 074...,

As for positive powers, the coefficients in boldface are also Fibonacci numbers, only that their signs alternate (this is the extension of the Fibonacci sequence to negative numbers) and also that this time the unit coefficients are one position ahead in the sequence relative to the φ coefficients, F(n+1). Thus, in general:

(− φ)− n = − F(n) φ + F(n + 1).   (f)

In the limit of very large values of n φ−n converges to zero and

φ = F(n + 1) / F(n).

This convergence of the ratio of two successive terms of the Fibonacci sequence to φ happens for other sequences where each term is the sum of the two previous ones. We'll now mention a specific example.

If you look carefully at the numerical values above, you'll see that, apart from the integer part, some decimal sequences reoccur, those of the odd positive and negative powers of φ. Adding equations (e) and (f), one finds:

φn + (− φ)− n = F(n − 1) + F(n + 1) = L(n),

where L(n) is a special integer, a Lucas number. From this, we see that after φ−n becomes smaller than 0.5 (i.e., for n > 1), all φn round to Lucas numbers and that the difference, being proportional to the inverse of φn, quickly becomes extremely small.

***

We now calculate the lengths of the other struts. For this, we will apply the Pythagorean theorem to estimate the diagonals of a sequence of rectangles. These constructions mean that these line segments can be built with a compass and straightedge, which means that their lengths are constructible numbers.


Fig. 2.2a: A Square built with B1 struts. Two HG1 struts represent the diagonal, they add to the length of a G1 strut.


Fig. 2.2b: A Rectangle built with B1 and G1 struts, the latter was constructed in the previous Figure.

Mathematically, this process can go on forever, obtaining the lengths of all square roots, and building the Spiral of Teodorus. However, with Zometool, we stop here.


The right angles in Fig. 2.2a have two B1 struts as sides, both with length 1. Thus, the length of the the diagonal (the hypotenuse of two right angle triangles) is given by h = √ (1 + 1) = 1.414 213 562..., a famously irrational number. This is the length of a G1 strut. The two HG1 struts shown have half this length, h/2 = √2 / 2 = 0.707 106 781... For other HG sizes, multiply by the different powers of φ.

The right angles in Fig. 2.2b have B1 and G1 struts as sides, with lengths of 1 and √2. Thus, h = √ (1 + 2) = 1.733 050 807... The two Y1 struts shown have h/2 = √3 / 2 = 0.866 025 403... For other Y sizes, multiply by the different powers of φ.

Because the diagonal is represented by yellow struts, I call this a Yellow rectangle. This has the proportions of A4 paper, which are kept when cut or folded in half withdways.

We now continue with the Golden rectangle. In Figure 2.2c, we construct it from a Square.


Fig. 2.2c: To build a Golden rectangle, we start with a Square, with a unit side (B1), on the left. We then mark the middle of the lower edge, open the compass to the upper right corner, and then mark that distance on the horizontal axis. The width of the full figure is now φ, in this case the length of a B2 strut; the length added to the side of the original Square is 1 / φ.

From this Figure, we can see that one of the properties of Golden rectangles is that subtracting a Square (on the left), we obtain a smaller, vertical Golden rectangle (on the right).


Fig. 2.2d: The diagonal of a Golden rectangle can be covered with two R1 struts.


Fig. 2.2e: A Rectangle built with B0 and B2 struts. Two Y1 sruts represent the diagonal.


The right angles in Fig. 2.2d have B1 and B2 struts as sides, with lengths of 1 and φ. Thus, h = √(1 + φ2) = √ ((5 + √5) / 2) = 1.902 113 032... The two R1 struts shown have h/2 = √ ((5 + √5) / 8) = 0.951 056 516... For other R sizes, multiply by the different powers of φ.

Finally, the right angles in Fig. 2.2e have B0 and B2 struts as sides, with lengths of 1/φ and φ. Thus:

h = √( 1 / φ2 + φ2)

Using equations (c) and (d), we obtain:

h = √( (φ − 1)2 + φ + 1 ) =
√( φ2 − 2φ + 1 + φ + 1 ) =
√( φ + 1 − 2φ + 1 + φ + 1 ) = √ 3, (g)

which is the same as in Fig. 2.2b. Thus the diagonal can also be covered by two Y1 struts, as we see in Fig. 2.2e! For this reason, I call this the Long yellow rectangle. Its area is given by φ × 1/φ = 1, i.e., the same as the Square in Fig. 2.2a.

These lengths will be useful through the whole site, we will use them below to study the regular polygons. For many more models explaining the properties of Zometool, see Hart (2000).

Zomable Polygons

 There are many types of two-dimensional figures. In what follows, we will concentrate on polygons. A polygon is a finite region of 2-D Euclidean space* bound by at least 3 straight line segments, or sides; these sides meet in pairs, in angles different from 0 and 180 degrees, at an identical number of vertices. These surface elements (henceforth "elements") - sides and vertices - are, together with the inner surface, also part of the polygon.

* This is the traditional definition of a polygon. We will not consider here apeirogons, skew polygons, or complex polygons.

A regular polygon (henceforth Polygons) is isogonal and isotoxal. This means that not only are the elements identical (sides of the same length, edge-vertex-edge angles - the "inner angles" - identical), but the polygon looks the same from all elements of a particular type. It follows from this that the polygon has a well defined centre and that all elements of a particular type are equidistant from it.

The names of polygons generally denote the number of sides, which is the the same as the number of vertices; for instance the name ``pentagon'' means ``five sides'' in Greek. In what follows, we will use capitalised polygon names to refer to their regular forms. The non-Greek names are the triangle, which refers to the three inner angles, and the quadrilaterals: Isogonal quadrilaterals are called rectangles, isotoxal quadrilaterals are called rhombuses; a regular quadrilateral (isogonal and isotoxal) has a special name, the Square.

We have shown above some special rectangles that can be built with Zometool, i.e., that are Zomable. We now expand this, including regular polygons and special rhombuses. Because Zometool represents only points (with the balls) and line segments (with the struts), it only represents the surface elements of the polygons, the vertices and the sides. This study will be useful for the latter study of figures in higher dimensions. We also display a strut connecting the vertex of the polygon to its center if this can be represented with Zometool, which tells us something about its metric properties.


Fig. 2.3a: Triangles can be huilt with B and G struts.


Fig. 2.3b: Squares, which we have already seen built with B struts (fig. 2.2a), can also be built with G struts. Additionally, we can also see that the distance from the centre of the square to a vertex can be represented by struts of the other colour.


Fig. 2.3c: The Pentagon can only be built with B struts.


Fig. 2.3d: Hexagons can be built with B and G struts; this is a consequence of this also being true for Triangles. Additionally, we can also see that the distance from the centre of the square to a vertex can be represented by struts of the same colour and length. This means that the distance of the centre to the vertex of an Hexagon is the same as its side, i.e., the Hexagon is radially equilateral. It is the only Polygon with this characteristic.


With Zometool, we cannot build Heptagons, Octagons and Nonagons. The fact that we cannot build Octagons with Zometool will have important implications later. However, we can build one last convex Polygon, in the Figure below.


Fig. 2.3e: Like the Pentagon, the Decagon can only be built with B struts. Additionally, we can also see that the distance from the centre of the square to a vertex can be represented by φ times larger B struts.


As discussed below, we can build additional non-convex Polygons: the Star polygons. We will also discuss several special non-regular quadrilaterals.

Geometric relations

We will now discuss several geometric transformations between polygons, which will be useful to understand all polytopes in general.

Duality: If polygons A and B are dual, then to a vertex of A corresponds an edge of B. This is a reciprocal relation, so to each vertex of B corresponds an edge of A.

We now show examples of the rectangle-rhombus duality: To the same angles of the Rectangles correspond the same side lengths of the rhombuses; to the alternate side lengths of the rectangles correspond the alternate angles of the rhombuses. The rhombuses touch the rectangles at right angles with a line from the vertex of the rectangle to the centre; this is known as the `` Dorman Luke'' construction, which can be used in any case where a polygon has a centre. In the figues below, we demonstrate this by drawing a special circle, which touches all the vertices of the rectangles (i.e, this circle circumscribes the rectangles) and tangentially touches all the sides of the rhombuses (i.e., this circle is inscribed in the rhombuses).


Fig. 2.4a: The Yellow rectangle and its dual, the Yellow rhombus. To achieve this particular construction, we had to glue two yellow rectangles by their longer sides, obtaining a larger Yellow rectangle.


Fig. 2.4b: The Golden rectangle and its dual, the Golden rhombus.


Fig. 2.4c: The Long yellow rectangle and its dual, the Long yellow rhombus.


Thus, the duals of the Yellow, Golden and Long Yellow rectangles in Figs. 2.2b, d and e are the Yellow, Golden and Long yellow rhombuses in Figs. 2.4a, b and c.

If a polygon is regular, its dual is of the same kind, but rotated by 180 deg / n, where n is the number of sides of the Polygons (see Fig. 2.4d). Thus, unlike most other types of polygons, regular polygons are self-dual.


Fig. 2.4d: The dual of a regular Polygon like the Pentagon is of the same kind: another Pentagon. The inner and outer Pentagons are duals of each other.


Rectification: Rectification consists of marking the mid-points of the sides of a polygon and cutting off the vertices at those points. This results in a new polygon, which is the rectification of the first one.

In the case of a Polygon, the rectification is of the same kind, but rotated by 180 deg / n, where n is the number of sides of the Polygons, and smaller. In Fig. 2.4d, the inner Pentagon, which is the dual of the outer pentagon, is also its rectification.

However, the rectification is not the same as a duality. If we rectify Yellow, Golden or Long yellow rhombuses, we obtain again Yellow, Golden or Long yellow rectangles, but in an orthogonal direction to that in Figs. 2.4a, b and c. In the next Figures, we see that the Rhombic diagonals are in the same proportions as the sides of the resulting rectangles, but twice as long (thus, all struts of the rectangles in Figs. 2.5a, b and c are necessary to trace the diagonals of their respective rhombuses).


Fig. 2.5a: The rectification of a yellow rhombus is a Yellow rectangle.


Fig. 2.5b: The rectification of a Golden rhombus is a Golden rectangle.


Fig. 2.5c: The rectification of a Long yellow rhombus is a Long yellow rectangle.


Note that the sides of the rhombuses are as long as the diagonals of the rectangles. Also, it should be clear that the rectifications of these Rectangles results again in the original rhombuses, but with half the size.

The rectifications of Figs. 2.5a, b and c are special cases of rectifications of quadrilaterals, which by Varignon's theorem are always parallelograms.

Stellation and faceting: A stellation extends the edges of a polygon until they meet other similarly extended edges of the same polygon. Stellations can transform a Polygon into a larger star polygon. The simplest case is that of the Pentagon (if we stellate Triangles and Squares, their edges would never meet), which stellates into a Pentagram (see Figure).

Facetings cut into a polygon, but preserve its vertices. Like the stellations, they cannot be convex. In Fig. 2.6b, we see show an example of faceting: the Pentagram is a faceting of a larger Pentagon, which is the dual of the inner one.


Fig. 2.6a: By extending the edges of the inner Pentagon, with vertices in white and dark blue B1 struts, the stellation operation results in a Pentagram, with vertices in black. The new side is 1 + 2 φ = φ3 larger than the side of the Pentagon. The inner white vertices are false vertices of the Pentagram: two sides intersect there, but none ends there.

The new Triangular areas that were added to the Pentagons are called Golden triangles. Studying them we can see why all edges can be represented by blue struts.


Fig. 2.6b: The Pentagram can be inscribed in a larger Pentagon, also with black vertices and B3 struts, i.e., φ2 larger than the side of the inner Pentagon. As we can see, this is the dual of the inner Pentagon. The Pentagram is a faceting of this larger Pentagon.

The new Triangular areas that were added to the Pentagram in this figure are Golden gnomons, which have the property that, when added to a Golden triangle, they make a φ times larger Golden triangle. In this Figure, there are 15 Golden Gnomons and 20 Golden triangles.


In the next figure, we stellate a Hexagon. This results is a regular compound of two Triangles with sides three times larger, the Hexagram. This is the simplest of the regular polygon compounds, the polygrams. This Hexagram can be inscribed in a larger dual Hexagon, which it facets.


Fig. 2.7: The stellation transforms the inner Hexagon, with vertices in white and light green G1 struts, into a regular polygon compound, the Hexagram, with vertices in black. The inner white vertices are false vertices of the Hexagram.


The larger the number of sides/vertices of a polygon, the larger the number of distinct stellations and facetings. This is, as we'll see now, the case of the Decagon, for which we show all three distinct stellations. The first is a Polygram, consisting of two Pentagons. The second is a regular star polygon, the Decagram. The third is another Polygram, a compound of two Pentagrams. All of these are also facetings of of the Decagon.


Fig. 2.8a: The first stellation of the Decagon (here in dark blue B1 struts and white vertices) is a compound of two Pentagons, here with yellow vertices. The white vertices further in are now false vertices.


2.8b: The second stellation of the Decagon is the Decagram, here with black vertices. The white and yellow vertices further in are now false vertices. The side of the decagram is now represented by three B1 and two B0 struts, which means it has a length 3 + 2 / φ = 1 + 2 φ = φ3 larger than the side of the inner Decagon. This is the same proportion as for the Pentagram and Pentagon.


Fig. 2.8c: The Decagram can be inscribed in a Decagon, which has edges built with B2 struts. This outer Decagon is therefore φ times larger than the inner Decagon. Thus, the distance of a vertex to the center, which was φ, now becomes φ + 1 = φ2. The Decagram is a faceting of this larger Decagon.

Notice how the radial struts bissect the quadrilaterals into Golden triangles, they also define central Golden triangles. Notice too the two sizes of Golden gnomons.


Fig. 2.8d: The third and last stellation of the Decagon is a compound of two Pentagrams, here with red vertices. These also result from stellating the two Pentagons with yellow vertices. The white, yellow and black vertices further in are now false vertices. Figures 2.8c and d show how many times φ appears in these geometric objects.


Isomorphism: The Pentagon and Pentagram are isomorphic: they have same number of sides and vertices with the same topology (a single circuit going through five vertices and sides). They have the same configuration matrix. The same happens for the Decagon and Decagram. However, the Hexagon and the Hexagram are not isomorphic, since in the latter case there is not a single circuit that cover all vertices, but two (the two Triangles of the compound) - the two figures have different topologies.

Metric properties and symmetry

 In this page, I will not dwell much on the mathematical aspects of polytopes; the references cover that very well. However, a few of the metric properties of polygons keep reappearing and being useful for understanding the polytopes in this page. For this reason I will discuss them here. I will however, make a couple general remarks about their symmetries first.

Some polygons have central symmetry: each element is reflected through the centre of the polygon into an identical one. In such polygons the number of sides / vertices must be even, as for instance, in the aforementioned rectangles and rhombuses. This will be important for understanding rectified polytopes.

The regular polygons are much more symmetric, having dihedral symmetry. If their number of sides is even, then they also have central symmetry, which is in that case a sub-symmetry within the dihedral symmetry.

The idea of a polygon as a circuit mentioned above will be discussed in more detail now. As we go around a polygon, we pass through n vertices, while completing p turns around the centre. For convex Polygons, p =1, but for the Pentagram, we actually complete two turns around the centre, and for a Decagram 3 turns. This number is known as the polygon's density.

Note that if there are common factors in n and p, we are in the presence of a Polygram. For instance, in Fig. 2.7, n = 6 and p = 2, hence two is a common factor, and thus we have 2 independent circuits (Polygons), each with n / p = 3 sides. Equally, for Fig. 2.8a, n = 10 and p = 2; 2 is the common factor and because of this we have 2 Polygons, each with n / p = 5 sides. Finally, in Fig. 2.8d, n = 10 and p = 4, 2 is a common factor, so we have 2 Polygons, each with n = 5 and p = 2, i.e., two Pentagrams.

The angle subtended by two vertices as seen from the centre must add to 360 degrees × p:

α = 360 p / n,

This angle is also the change of direction at each vertex, as we move from one side to the next. Thus, the inner angle at the vertex is the supplementary angle:

β = 180 − 360 p/n.

These angles are listed in Table 1 below, for instance, for a Triangle, the inner angles are 60 degrees. The sum is therefore 180 degrees. This sum is valid for all triangles in flat, Euclidean spaces.




Fig. 2.9: Definition of the main angles of a Polygon. The black circle circunscribes the Polygon, and has radius R0, the side is given by ℓ.


Now, if the vertices are at distance R0 from the centre (see Fig. 2.9), the sides (with length ℓ) are the chord of α times R0:

ℓ = R0 chord(α)

Replacing the sine expression of the chord and solving for R0, we obtain (see also Fig. 2.9):

R0 = ℓ / (2 sin (α / 2)) = ℓ / ( 2 sin (180 p / n))

The minimum distance of the side to the centre is given by:

R1 = ℓ / (2 tan (α / 2)) = ℓ / (2 tan (180 p / n))

The surface of the Polygon is the same as that of 2 n triangles with base ℓ/2 and height R1.

Another important quantity is chord(β). This is, for any particular vertex P, the distance between the two vertices it connects to divided by their distance from P (ℓ). This gives us side lengths of some facetings, as seen for the outer Pentagon in Fig. 2.6b: the side of the Pentagram is the side of the outer Pentagon times chord(β).

The Zomable Polygons are constructible, which is synonymous with the sines in the equations for R0, R1 and chord(β) being a subset of the constructible numbers, the exact trigonometric values. From these, we obtain the values listed below:

Polygon α β R0/ℓ R1/ℓ chord(β)
Triangle 120 60 1 / √3 1 / (2 √3) 1
Square 90 90 √2 / 2 1 / 2 √2 / 2
Hexagon 60 120 1 √3 / 2 √3
Pentagon 72 108 √( (5 + √5) / 10) 1 / (2 √(5 − 2 √5)) (√5 + 1) / 2
Decagon 36 144 (√5 + 1) / 2 √(5 + 2 √5) / 2 √((5 + √5) / 2)
Pentagram 144 36 √2 / √(5 + √5) 1 / (2 √(5 + 2 √5) ) (√5 − 1) / 2
Decagram 108 72 (√5 − 1) / 2 √5 / (2 √(5 + 2 √5)) √2 √(5 − √5) / 2
Metric properties of Zomable Polygons

The R0 lengths for the even-sided Polygons are indicated in boldface because they are Zomable: The Decagram in Fig. 2.8c results from stellating the Decagon in Fig. 2.3e. The distance to the centre of the Decagon is given by a B2 strut (length φ), the distance to the centre of the Decagram is given by a B2 and a B1 strut, with total length φ + 1 = φ2. This was not represented with a B3 strut because the vertices of the Decagon are in the way, which means that the vertices of the outer Decagram are ``above'' those of the Decagon, not in dual positions. As implied by the φ times larger distance from the centre, the vertex arrangement of the Decagram is φ times larger that for the vertices of the Decagon is stellates. This will be important for quite a few polytopes to be mentioned later!


Next: Polyhedra