Everyone loves a good shape. You may think that you learnt all the shapes at primary school, but there are plenty still around that mathematicians find interesting. Sprinkled through Issue 03 of Chalkdust were some of the team’s favourite shapes. Here we have collected them together, and added many more.

We’d really love to hear about yours! Send them to us at contact@chalkdustmagazine.com, tweet them to @chalkdustmag, or post them on Facebook and you might just see them on a future blog!

The triangle (Eleanor Doman)

My favourite shape is one you cannot get away from—the triangle. The cosine rule guarantees that for any triangle with the lengths of the sides given, there is a unique combination of internal angles. Simply put, the triangle is rigid provided the sides are fixed, making it an essential shape in the fields of architecture and engineering.

Möbius strip (Rob Beckett)

My favourite ‘shape’ is the one sided non-orientable surface called the Möbius strip. This can be created by simply twisting a long strip of paper and gluing the ends together. One of the explanations most regularly associated with the Möbius strip is that of MC Escher, who described an ant crawling along its surface. The ant would be able to do this and return to his starting point having not even crossed an edge (or maybe it keeps crawling on indefinitely hoping to find the end!).

In the Numberphile video Möbius bridges and buildings Carlo H Séquin (UC Berkeley) considers using the idea of a Möbius strip to create aesthetic bridges and buildings.

Gabriel’s horn (Matthew Scroggs)

My favourite shape has a finite volume but an infinite surface area. This means that it is possible to fill it with paint, but not possible to paint its surface.

Gabriel’s horn can be created by taking the curve $y=\frac1x$ (for $x\geq1$) and rotating it about the x-axis. Its volume is
\begin{align*}
\pi\int_1^\infty y^2\,dx&=\pi\int_1^\infty \tfrac1{x^2}\,dx\\&=\pi\left[-\tfrac1x\right]_1^\infty\\&=\pi\lim_{k\to\infty}(1-\tfrac1k)\\&=\pi,
\end{align*}
which is finite. Its surface area is
\begin{align*}
\int_1^\infty y\sqrt{1+\left(\tfrac{dy}{dx}\right)^2}\,dx&=\int_1^\infty \tfrac1x\sqrt{1+\tfrac1{x^4}}\,dx.
\end{align*}
Whenever $x$ is positive, $\displaystyle\tfrac1x\sqrt{1+\tfrac1{x^4}}$ is greater than $\tfrac1x$ and so
\begin{align*}
\int_1^\infty \tfrac1x\sqrt{1+\tfrac1{x^4}}\,dx\geq&\int_1^\infty \tfrac1x\,dx\\&=\left[\ln x\right]_1^\infty\\&=\lim_{k\to\infty}(\ln k),
\end{align*}
which is infinite.

The circle (Pietro Servini)

Mathematically, the circle is the set of all points in a plane that are at the same distance (the radius) from the centre and it’s intimately related to the most famous irrational number, $\pi$, which appears in the most unexpected places.  Historically, the circle inspired the creation of the wheel, generally accepted as mankind’s greatest invention, giving us the ability to move easily and at speed and, later, develop complex machinery relying on gears and cogs.  The first known wheel, however, was not used for locomotion but for pottery: a potter’s wheel was found in Mesopotamia (modern day Iraq) dating to around 3,500 BC, relatively late in humanity’s development. Perhaps most importantly, the circle then gives rise to the 3D sphere, whose aerodynamic characteristics and ability to roll have spawned so many sports without which the world would be so much bleaker…

4-simplex  (Belgin Seymenoğlu)

We can start with just a point which, believe it or not, is already a simplex. Then if we introduce a second point, we can connect the two to get a new shape called a 1-simplex (or a line to you and me). Next, if we take a third point, and connect it to our two other points, we have the 2-simplex, otherwise known as a triangle. But if we then connect our three points in the triangle to yet another new point, we get a three-dimensional shape: the tetrahedron (or the 3-simplex.

What’s more, there is yet another member of the family: a four-dimensional shape. This shape is called the 4-simplex, and it has five vertices. The 4-simplex is useful in population biology because if you have, for example, five different species, you can represent the fractions of each population by plotting a point in the 4-simplex.

If that’s not enough for you, you can make a five-dimensional, six-dimensional or even an $n$-dimensional simplex!

Penrose tiles (Rudolf Kohulák)

My favourite shape is a rhombus that has been split into two pieces called ‘kite’ and ‘dart’. These shapes might not look interesting, but the British physicist Roger Penrose discovered an unusual feature of these objects. They can be arranged to cover the whole plane without any gaps or overlaps. However, the resulting image is highly unsymmetrical. For instance, it lacks translational symmetry (ie you cannot shift the pattern such that the result would end up being identical to the original picture). The discovery revolutionised the field of crystallography and led to the identification of quasicrystals.

Sierpinski triangle (Nikoleta Kalaydzhieva)

My favourite shape is the Sierpinski triangle. It is one of the most basic fractal shapes, but appears in various mathematical areas. What I find fascinating about it is how many different ways there are for constructing it. For example, you could use a methodical geometric approach by inscribing a similar triangle in the original one via its midpoints and iterating. Another, more intriguing construction, is via the Chaos game. You can even construct it using basic algebra, by shading the odd numbers in Pascal’s triangle.

Helicoid (Alexander Doak)

My favourite shape is the helicoid, as it has many interesting geometric properties. Firstly, it is a ruled surface. The helicoid is constructed by moving a straight line in space; in this case by rotating it about an axis while moving along said axis at a constant speed. Ruled surfaces are very popular in architecture, such as hyperboloid cooling towers and, of course, helicoid staircases. Secondly, it is a minimal surface. In fact, it has been proven that the helicoid, along with the plane, are the only ruled minimal surfaces!

My favourite geometrical figure is the heptadecagon, a regular polygon with 17 sides. It comes with the history of a great challenge that required the efforts of almost eighty generations of mathematicians to solve. Ancient Greeks knew how to construct polygons with 3, 4, 5, 6, 8, 10, 12, 15, 16, and 20 edges using only a straightedge and compass, while 18th century algebraists knew that it was impossible to use the same tools to construct polygons with 7, 9, 11, 13, 14, 18 and 19 sides. Gauss, at 19, was the first to prove that the heptadecagon was constructible.

The light cone (Matthew Wright)

My favourite shape is the light cone. It is a four-dimensional shape lying in space-time, and it is the path travelled by beams of light emitted from a single point. Although a simple concept, it turns out to be of fundamental importance: it determines the entire notion of causality. Everything that can be causally affected by an event at one point in space and time must lie within that event’s light cone, since nothing can travel faster than the speed of light. Einstein realised that gravity wasn’t a force in the conventional sense, but rather distorts the structure of space and time, tipping and deforming the light cones in the process. This is why nothing can escape a black hole: the light cones are tipped over so much that everything in the future of the light cone must lie inside the black hole.

Inspired by the shapes above? Think you know better? Remember to send us your favourite shape either by email (contact@chalkdustmagazine.com), on Twitter (@chalkdustmag) or Facebook (/chalkdustmag).

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