GeoGebra Art

In the classes I teach I often use GeoGebra for demonstrations; I assign GeoGebra constructions as homework, and our senior math majors use it in their oral presentations. GeoGebra, if you don’t know, is a Java-based dynamic geometry program freely available on the web. GeoGebraTube has thousands of applets developed by members for use in the classroom.

Now, GeoGebra is used mainly by high school teachers, and most of the applets available online are developed with that in mind. But the emphasis on algebra and coordinates by state curricula means that the program’s possibilities are vastly curtailed in common practice.

In trying to explore the software’s capabilities, I’ve set myself the task of constructing some of the fractals described in Mandelbrot’s Fractal Geometry of Nature. Images of my first attempts are shown below. At this point I’m not going to go into what makes a fractal a fractal, contenting myself merely with the following quatrain, which comes from Jonathan Swift:

So, Nat’ralists observe, a Flea,
Hath smaller Fleas that on him prey,
And these have smaller Fleas to bit ’em,
And so proceed ad infinitum.

Click on an image to view the slideshow and read the captions. Interactive versions can be found by going to my GeoGebra profile page. I won’t expand on my cleverness in the succinct programming that produced these images, but some of them are extremely clever.


Rotational Ornaments

An isometry is a transformation of the plane that takes each line segment to a congruent line segment and each angle to a congruent angle. Basic types of isometries include reflections, rotations, and translations.

In the last last post, we saw that a symmetry of a geometrical figure is an isometry that preserves the figure. A group is a collection of symmetries satisfying: (1) if a symmetry T is in the group, then its inverse is also in the group; and (2) if T and U are in the group, then the composition U ∘ T is also in the group.

We discussed the dihedral groups D3 of the equilateral triangle and D4 of the square. Now consider the set of symmetries that preserves the regular pentagon.

pentagon symmetry 1

In this case we have five reflections: R1, R2, R3, R4, and R5. In addition, we have clockwise rotation S through one fifth of a complete revolution, or 72°, and its compositions with itself: S2, S3, S4. So the dihedral group D5 consists of ten symmetries: I, S, S2, S3, S4, R1, R2, R3, R4, and R5. Once again, we can write D5 = 〈R1,R2〉, where R1 and R2 are reflections across adjacent lines making an angle of 36° with one another. For instance, R1 ∘ R2 is a 72°-rotation.

Now imagine extending the sides of the pentagon as shown.

pentagon symmetry 2

This “breaks” the symmetry. No longer can we reflect. However, the rotations still preserve the figure. So the symmetry group now consists of five symmetries: I, S, S2, S3, and S4. This is referred to as the cyclic group C5. It’s generated by S, so we can write C5 = 〈S〉. The cyclic group is a subgroup of the dihedral group.

In general, the cyclic group Cn is generated by a rotation through 360°/n, or one nth of a revolution, while the dihedral group Dn is generated by two reflections whose lines make an angle of 360°/2n. Cyclic and dihedral groups abound in ornamental symmetry; Owen Jones’ 1910 Grammar of Ornament contains a wealth of examples.

The group D1 contains a single reflection; it may be thought of as the set of symmetries of the letter V. This is bilateral symmetry, as exhibited in the following Egyptian ornament.


The group D2 contains two reflections across perpendicular lines; it may be thought of as the set of symmetries of the letter H. We see examples of it in the following Greek and Pompeian ornaments.


Notice that if we ignore the color in the Pompeian ornament, then we have a D4 symmetry instead. A true example of D4 symmetry is given in the following Persian ornament.


Larger dihedral groups are rarer but still occur. The following ornaments are, respectively, Assyrian, Chinese, and Byzantine; they have symmetry groups D5, D8, and D22.

Dihedral other

Ornaments that exhibit only a cyclic symmetry generally have some kind of overlapping knot pattern or swirl element which prevents reflection. The cyclic group C1 is the trivial group consisting of just the identity.


The cyclic group C2 contains a single rotation through 180°. This may be thought of as the set of symmetries of the letter N. The following Celtic ornament has C2 symmetry; the overlapping of the knot prevents its having D2 symmetry.


The cyclic group C4 is generated by a 90°-rotation. Examples include the following medieval European and Moorish ornaments.


The group C6 is generated by a 60°-rotation, while C8 is generated by a 45°-rotation; these are exhibited in the following medieval European and Arabian ornaments.

Cyclic other

In the latter case, we have to ignore the box design around the circular ornament. If we don’t, then the symmetry group is C4 rather than C8. In general, Cn contains Cm as a subgroup if and only if m is a divisor of n. So C8 contains C1, C2, and C4. Similarly, C6 contains C1, C2, and C3. We’ll return to this in the next post on the topic.

Byrne’s Euclid

In 1847, an eccentric new edition of Euclid’s Elements was published in Britain. Designed by the otherwise obscure mathematician Oliver Byrne, it replaced letter variables with color diagrams and symbols “for the greater ease of learning.”

byrne title

History has tended not to agree with Byrne on the pedagogical success of his edition; the market apparently didn’t, either, for the edition didn’t sell well, and its extravagant expense sent the printing firm into bankruptcy.


Despite all of this, it is a true delight to read. A facsimile edition has been published, but a complete scan of the book is available online at this link. Byrne’s Euclid has been called “one of the oddest and most beautiful books of the century.” It was featured at the Great Exhibition of 1851, and has been seen as an anticipation of the Bauhaus school of design.

Fra Luca da Pacioli

The painting in the blog header is a detail from a portrait of Fra Luca da Pacioli (1466 – 1517), Italian Renaissance mathematician and Franciscan friar.


Pacioli is not known for any original discoveries. He’s most famous for his books, e.g., his Summa de Arithmetica, Geometria, Proportioni et Proportionalità (Summary of Arithmetic, Geometry, Proportion, and Proportionality), a textbook on mathematical knowledge at the time, and De Divina Proportione (On the Divine Proportion), a book on geometry. In the latter, he expounds on the golden ratio and its relationship to the Platonic solids. It was illustrated by his friend, Leonardo da Vinci:

pacioli polyhedron

The painting itself contains many interesting mathematical objects. The model hanging in the upper left-hand corner is a truncated rhombicuboctahedron, an Archimedean solid, i.e., a polyhedron whose vertices are all congruent to one another and whose faces are all regular polygons.

pacioli 4

In this case, the polygons are squares and equilateral triangles. Notice that the polygon is half-filled with water and that its panes show a reflection of the view through the window behind the painter.

Pacioli is making a demonstration out of Euclid on a piece of slate. A dodecahedron (a Platonic solid composed of regular pentagons, the construction of which is the climax of Euclid’s Elements), sits on the book to the right. The young man standing behind Pacioli has not been identified, but he’s conjectured to be Albrecht Dürer, a painter of the Northern Renaissance who spent some time in Italy and later wrote a book on geometry and perspective. Some of Dürer’s graphic works contain mathematical themes.

Mathematics is in many ways a very tradition-oriented subject. Any mathematician who receives a doctorate was advised in his or her dissertation by a senior mathematician in a master-pupil relationship. An advisor can have a profound influence over their students’ mathematical predilections, understanding, and outlook. The Math Genealogy Project allows one to trace any line of “ancestry” from pupil to master as far back as the record goes, or to start with a historical mathematician and trace their mathematical “progeny” down to modern times.

While reading about Pacioli’s life, I explored the various lines of his mathematical “descendants.” It is unknown who Pacioli’s advisor was; his sole student was a man named Domenico da Ferrara. Among Ferrara’s students was Nicolaus Copernicus, the great pioneer of modern astrophysics. I continued tracing this continuous line through the sixteenth, seventeenth, eighteenth, nineteenth, and twentieth centuries, meeting along the way such luminaries as Johann Pfaff, Carl Friederich Gauss, and Karl Weierstrass. And, much to my surprise, at the beginning of the twenty-first century, I arrived at…myself, your humble RGC math professor.

This doesn’t make me anything special, however, for Pacioli has had more than a hundred thousand descendants over the centuries!