The Regular Enneagon, and Three Regular Enneagrams

enneagon and enneagrams

The red figure above is a regular enneagon, or nine-sided polygon, and it has three regular enneagrams (or “star enneagons”) inside it. The light blue figure is called a {9,2} enneagram. The green figure can be viewed two ways: as a {9,3} enneagram, or as a compound of three equilateral triangles. Finally, the yellow figure is a {9,4} enneagram.

To see what these numbers in braces mean, just take a look at one of the yellow enneagram’s vertices, then follow one of the yellow segments to the next vertex it touches. Count the vertices which are skipped, and you’ll notice each yellow segment connects every fourth vertex, giving us the “4” in {9,4}. The “9” in {9,4} comes from the total number of vertices in this enneagram, as well as the total number of segments it has. The blue and green enneagrams are analogous to the yellow one. These pairs of numbers in braces are known as Schläfli symbols.

I should mention that some people call these figures “nonagons” and “nonagrams.” Both “ennea- and “nona-” refer to the number nine, but the latter prefix is derived from Latin, while the former is based on Greek. I prefer to use the Greek, since that is consistent with such Greek-derived words as “pentagon” and “hexagon.”

Finally, there is also an “enneagram of personality,” in popular culture, which some use for analyzing  people. Aside from this mention of it, that figure is not addressed here — nor is the nine-pointed star used as a symbol for the Bahá’í faith. However, it’s easy to find information on those things with Google-searches, for those who are interested.

A Rhombic Mandala Based on Pi Over Nine

ninthsThe interior angles in these rhombi all measure π/9 radians, or some whole-number multiple of that amount, up to 8π/9 radians.

A Forgotten Mandala, from 2010

Someone found this, and “liked” it, in my old Facebook pictures. I had forgotten all about it, until this happened. It is a mandala, made of rhombi, with nine-fold symmetry, made in 2010 with Geometer’s Sketchpad — two years before I started this blog.

from 2010

Nine (2015) / Nine (2013)

First, the newer version I just made:

nine 2015

Next, the 2013 version, which I recently found, along with a bunch of other previously-lost stuff from around then. The two are simply color-inversions of each other, according to the rules for color-inversion used by MS-Paint.

nine 2013

A Polyhedral Demonstration of the Fact That Nine Times Thirty Equals 270, Along with Its Interesting Dual

30 times 9 is 270

It would really be a pain to count the faces of this polyhedron, in order to verify that there are 270 of them. Fortunately, it isn’t necessary to do so. The polyhedron above is made of rhombus-shaped panels which correspond to the thirty faces of the rhombic triacontahedron. Each of these panels contains nine faces: one square, surrounded by eight triangles. Since (9)(30) = 270, it is therefore possible to see that this polyehdron has 270 faces, without actually going to the trouble to count them, one at a time.

The software I used to make this polyhedron may be found at http://www.software3d.com/Stella.php, and is called Stella 4d. With Stella 4d, a single mouse-click will let you see the dual of a polyhedron. Here’s the dual of the one above.

30 times 9 is 270 -- the dual

This polyhedron is unusual, in that it has faces with nine sides (enneagons, or nonagons), as well as fifteen sides (pentadecagons). However, these enneagons and pentadecagons aren’t regular — yet — but they will be in the next post.

Ten Enneagrams

Image

Ten Enneagrams

These enneagrams are of the {9/3} variety, which means each one is made of three concentric, equilateral triangles. One of these enneagrams is at the center; the other nine surround it.

Only Nine School Days Left This Year

Image

Only Nine School Days Left This Year

Due to an unusual amount of Winter weather this school year, the school year where I teach has been extended to June 6, creating what many are calling “the school year that will not end.” It will end, of course, but the already-long wait for Summer vacation is getting to many of us — students, parents, teachers, and administrators alike.

The countdown is now at nine school days left: four next week, and five the week after that. In honor of this point in the countdown, I created this image based on the number nine, using Geometer’s Sketchpad and MS-Paint.

Changing the “Nine” in the Nine-Point Circle

Image

Constructing the nine-point circle is an interesting exercise in geometry. In the above triangle ABC, the segments inside the triangle are its three altitudes, with the “feet” of the altitudes labeled E, F, and G. The midpoints of the sides of the triangle are labeled L, M, and N. The orthocenter, where the three altitudes meet, is labeled O, and then the midpoints of the three segments connecting the orthocenter to each of the triangle’s vertices are labeled X, Y, and Z.

It has been long proven that these three sets of three points each (E, F, G; L, M, N; and X, Y, Z) lie on the same circle, for any triangle. Point Q is at the center of this nine-point circle.

The diagram above uses, as triangle ABC, a triangle which is both acute and scalene — and in such a triangle, the nine points in question are in nine different locations. Of course, triangles do not have to be acute and scalene — and for some other types of triangle, the nine points end up in fewer than nine distinct locations.

9PC obtuse scalene

Of other classifications of triangle, only this one, an obtuse and scalene triangle, still has the nine points in nine different locations. With other triangles, the number of such locations decreases.

As a next step, consider a triangle which is acute and isosceles:

9pc acute isos 8

The base of this isosceles triangle is segment AB, and it is on segment AB that two of the nine points end up in the same place. Point G, the foot of the altitude to the base, is at the same place as point N, the midpoint of the base. Since the other seven points remain distinct, this type of triangle has its nine points in eight locations.

9PC obtuse isosceles 8

Another triangle which has eight distinct “nine-point circle” points is the obtuse, isosceles triangle, for the same reason:  the foot of the altitude to the base (G) and the midpoint of the base (N) are in the same place. Eight is not the limit, though — this number can be reduced still further. As one attempts to do so, it doesn’t take long to figure out that there is no way to reduce this number to seven . . . but six is possible:

9pc equilateral 6

For the nine points under examination to end up in only six distinct locations, as seen immediately above, a triangle is needed which is equilateral (and equiangular as well, for you can’t have one without the other when dealing with triangles). In such a triangle, each side-midpoint ends up at the same place as an altitude-foot, providing three of the distinct six points. The other three are the midpoints of the segments connecting the orthocenter to each vertex. Also, it is only for this type of triangle that the orthocenter (O) is the center of the nine-point circle itself (Q). One might think that this type of triangle, being regular, would minimize the number of distinct locations for the “nine” points . . . but that is not the case.

9pc right scalene 5

To reduce this number below six, right triangles are needed. With a scalene right triangle, there end up being five such locations:  the midpoint of each side, the vertex of the right angle, and the foot of the altitude to the hypotenuse. However, five is not quite the minimum.

9pc right isosceles 4

If a right triangle is isosceles, rather than scalene, the foot of the altitude to the hypotenuse moves to the midpoint of the hypotenuse, and this reduces the number of distinct “nine-point circle” points to its absolute minimum:  four. Such a triangle is also called, of course, a 45-45-90 triangle. Interestingly, these four points may be used as the vertices of a square (not shown in the diagram above) which has an area exactly one-half that of triangle ABC. The proof of this is left as an exercise for the reader.

{Later edit, March 2018:  an alert reader pointed out to me that I “missed some obtuse [triangles] that have only eight or six points on the nine-point circle.” Good catch, F.D.!}