Mathematical Names for Objects Quiz

It's not quite a perfect cone - you can see that the tip is a bit rounded, the sloping face has an uneven surface, and there are ridges as we move from sugar cone to ice cream to chocolate - but pretty recognisable as a cone. It's even called an ice cream cone!

To be more precise, if this were an ideal example, it would be called a right circular cone. That means that the base (on which it would stand if you set it tip-upwards) is a circle, and a line from the tip (called the apex) to the centre of the base makes a right angle when it makes contact. A cone can have other curves as a base, such as an ellipse, and the apex can look as if it has been pushed sideways so that the altitude does not hit the base at a right angle. This is called an oblique cone. Depending on the context, the term cone may refer to the surface formed when every point on the base is smoothly connected to the apex, or it may refer to a solid shape for which that is the exterior surface.

Once again, there are surface imperfections, so the sloping sides that should be perfectly flat have irregularities. But this structure is clearly in line with the mental image most people have when they hear the word pyramid. It has a polygon (a shape made up of straight edges) as a base, unlike the smooth curve of a cone. Like a cone, every point on the base is connected to the apex, which produces distinct faces, due to the angles formed at the vertices of the polygon, each of which is a triangle. Like cones, pyramids can be either right or oblique - this one is a right pyramid, with the apex directly above the centre of symmetry of the base.

Pyramids are given a more complete name by identifying the shape of the polygon on its base. This one is a rectangle - four straight sides that meet at right angles. It is probably a square, with all four of those sides the same length, but I cannot be sure of that from a picture. If the base had twenty sides (don't ask me why), it would be called an icosahedral pyramid.

Because this is a closeup of part of the surface of a soccer ball (football), the shape is not flat, as it should be; rather, it bends slightly down towards the stitching around each edge. But let's pretend it is flat. Because the shape is made up of straight sides that meet at sharp angles, it is an example of a polygon (from Greek roots meaning many angles).

Polygons are named according to the number of sides - for small numbers, this is usually a prefix based on Greek or Latin words for the number, followed by -gon. Exceptions include the three-sided polygon (triangle) and the four-sided polygon (quadrilateral). Once the numbers get large, it is more usual to describe it with a number: with 50 sides, it would be a 50-gon. The figure in the diagram has five sides, so it is called a pentagon. (Those who are familiar with the entire surface of a soccer ball, or who recognise the polygons which join on at each side may know that these are parts of hexagons, but they do not form the primary image, and are not completely seen.)

There are other words that are commonly used to describe a polygon, especially when the sides are all the same length, and meet each other at vertices which are the same size. These are called regular polygons, so this is a regular pentagon.

A tetrahedron (whose name means four bases) is a pyramid whose base is a triangle, so it can also be called a triangular pyramid. The image shows a regular tetrahedron, in which all four sides are equilateral triangles, so it looks the same no matter which one is chosen as the base.

A tetrahedron is the simplest example of a polyhedron. A regular tetrahedron is the smallest of the Platonic solids (named after the Greek philosopher), whose faces are all regular polygons. There are only five of these: tetrahedron (four triangular faces), cube (six square faces), octahedron (eight triangular faces - like two square-based pyramids joined at a common base), dodecahedron (twelve pentagonal faces) and icosahedron (twenty triangular faces). It is also the smallest deltahedron, a group of convex polyhedra (meaning their vertices all point outwards away from the centre of the shape) with all faces composed of equilateral triangles.

This is a Rubik's cube, a puzzle developed in the 1980s composed of 21 pieces (a central rotating mechanism to which the centre square on each face is attached, and 20 pieces that form the external set of squares on each face. These can then be rotated to produce an amazing number of different arrangements of the 54 squares on the surface. The challenge is to mix them up, then restore them so each side has a single color. (If that is too simple for you, more advanced challenges provide a particular arrangement which you can try to produce.) While it is clearly not a perfect cube, with each face divided into 9 separate moving pieces, it does have the overall shape of a cube: it has six faces which are all squares.

As mentioned in the discussion of tetrahedra, the cube is a Platonic solid. They are said to be tessellating, because you can fill a space with them, by stacking them neatly against each other. Of course, some trimming may be needed at the edges, but mathematicians don't worry about those little details of construction!

This diagram is useful as a reference in describing some of the terminology of polyhedra. They have flat surfaces called faces. Faces (F) meet along straight lines called edges. Edges (E) meet at corners called vertices (V). Using these three numbers to define the polyhedron allows mathematicians to perform analysis of all sorts of theoretical objects called polytopes, which are not confined to existence in our three-dimensional world, but which can be extended to have any number of dimensions. For a simple (real) convex polyhedron, the relationship between these numbers is given by Euler's rule: V - E + F = 2. The number changes for more complex polytopes. In the case of our cube, which has 8 vertices, 12 edges and 6 faces, we can see that 8-12+6 does indeed give 2.

This closeup of honeycomb in cross section shows that it is composed of shapes that have six sides, making them hexagons. Once again, the sides are not perfectly straight and uniform; nor are they lines which are only a single point wide, but in the real world lines just have to be very narrow, not microscopic! The sides of each hexagon are slightly thickened at the vertices, allowing them to join firmly on to their neighbours, and giving the opening a slightly curved appearance.

This diagram shows the hexagon's property of tessellating. This means that they can be laid down and joined in such a way as to completely cover a flat surface without any gaps. Actually, it's more rigorous than that. A regular tessellation must involve only one shape, and they must meet at vertices. The meaning of that can be seen if you think of the way bricks are laid (looked at from the side, so we are considering only that surface, not the solid space). They are offset so that the rectangles do not line up in adjacent layers. This means that the vertices of each layer are touching the sides of the next layer. Not a tessellation, even though there are no gaps (and it is much stronger than a tessellating arrangement would be, but that's another consideration).

Not many regular shapes tessellate on their own: equilateral triangles, squares, hexagons. That's it. For shapes to tessellate, you have to be able to join their vertices to form a complete 360-degree angle, and these are the only regular polygons for whom the interior angles at each vertex are factors of 360. For equilateral triangles, the angle is 60 degrees, so six of them meet at each vertex. For a square, it is 90 degrees, and four meet at each vertex; for a hexagon, it is 120 degrees, and three meet at each vertex.

Almost any kind of shape can be made to cover a surface smoothly. Once you start using irregular shapes (Escher's ducks spring to mind) or multiple geometrical shapes that fit together, the sky's the limit.

A prism is a three-dimensional figure with two parallel and identical faces (the red triangles in this diagram) whose edges are connected by parallelograms. If the two bases have not been moved sideways relative to each other, the faces will be rectangles, and it is a right prism. Just as the apex of a pyramid can be 'pushed' sideways, so the two bases of a prism can be slid sideways while remaining parallel to each other. The resulting prism will have faces that are still parallelograms (since they have two pairs of parallel sides), but the angles at each corner will no longer be 90 degrees, so they are not rectangles. You may be able to imagine this if you think about holding the two triangles above in your hands, then pushing one end away from you, and watching the faces stretch accordingly.

Like pyramids, prisms are often described by giving the shape of those faces - so this is a triangular prism. Indeed, it is a right triangular prism. I wanted to use a Toblerone bar package to illustrate the shape, but was unable to find a suitable image. Another place you may have encountered a triangular prism is in a high school science class, where they can be used to show that white light is a mixture of colours - shining white light into a triangular prism appropriately will cause the emerging light to be split into a spectrum, displaying separately the same colours as those seen in a rainbow.

Although they are not actually visible in the image, the two ends of the box (which are essentially parallel to each other) are rectangles, possibly even squares. That makes it a rectangular prism - and since the sides are all four rectangles, it is a right rectangular prism. That's quite a mouthful, so calling it a cuboid is easier. The term cuboid suggests that it resembles a cube, but is not one.

A cube could be considered a rectangular prism with square bases, and the four faces also being squares. If you start with that then pull the two ends apart, stretching the faces into rectangles, you have a cuboid. Turn it around so two of the former faces become the ends you can grab to push or pull again, and they will also become rectangles. It is still a cuboid, as all six faces are rectangles of some sort. Now hold it so that one face is on top and one is facing you. Push the top edge of the one that is facing you away, so the rectangles on each end get distorted into parallelograms, and you no longer have a cuboid. This is called a parallelepiped, and is an example of an oblique (rather than right) prism.

I hope you weren't thrown by the triangular frame holding these pool balls together in preparation for the start of a game. Many of the balls used in sports are described as spheres, but these have a smoother surface than, say, a tennis ball or a golf ball. And soccer balls are not spheres at all, they are made up of a number of plane shapes assembled to make a spheroid - resembling a sphere, but not being one. It is most commonly a truncated icosahedron, meaning that a regular icosahedron (remember them from earlier discussion of Platonic solids?) has had all its vertices cut off, or truncated. What is left is a polyhedron that has 12 pentagons and 20 hexagons as its 32 faces. The pressure of the air inside the ball makes it bulge, so it is not a perfect example, but it illustrates the concept.

Back to the sphere. This is the three-dimensional shape that corresponds to a two-dimensional (or flat) circle. One way to describe how to form a sphere is to take a circle, hold it at two points on either end of a diameter, and spin it around that axis. Just as a circle can be defined as a collection of points on a plane which are all the same distance (called the radius of the circle) from a fixed point, so the sphere can be defined as all points in three-dimensional space which are the same distance from a central point. Since all spheres are exactly the same shape, and differ only in size, a sphere is completely described by stating its radius.

While the earth is not a perfect sphere (even ignoring mountains and such, its spin means that the diameter at the equator is slightly larger than the diameter connecting the poles, so it is often described as an oblate spheroid), terminology from the earth's globe is often used when talking about spheres: start by selecting two points that are exactly opposite each other, and join them with a line called the axis; name them the poles, and any circle on the sphere's surface that has the same diameter as the sphere and divides it into two equal hemispheres is a great circle. The great circle which is at right angles to the axis is the equator. On a sphere, the shortest distance between any two points on the surface is along the great circle that connects them.

This ball is used in the game of rugby. Please don't ask me to be more precise as to exactly which form of rugby is played by this Italian team, I am more concerned with its shape. If you recall that a sphere can be described as the shape made if a circle is spun around an axis, this shape is formed when an ellipse is treated the same way. If you pretend this image is a flat object, you can see the shape of that original ellipse. Calling it an ellipsoid makes it clear that you started with an ellipse. Some prefer to call them spheroids, emphasizing the fact that they can be considered as distorted spheres, just as the cuboid can be considered a distorted cube.

A line between the two ends of the ball is called its major axis; a line between the two points on the long curved sides where they are furthest apart is called its minor axis. If it's a perfect ellipsoid, and you cannot tell whether it should be considered to rest with the longer or the shorter axis parallel to the ground, then it is pretty thoroughly described by calling it an ellipsoid and giving the length of those two axes. More commonly, there is a reason to consider one axis to be "up" - in the case of the earth, for example, we usually consider the North Pole to be up, so the major axis is horizontal, and it is called an oblate ellipsoid. If, however, you consider your ellipsoid and think that the major axis should be pointing up, you would describe it as a prolate ellipsoid.

Now on to a shape you may well never have heard of, the catenary curve. Technically, the Gateway Arch in St Louis MO is a weighted catenary arch, because its thickness changes (in order to provide sufficient support at the base for the load it has to bear). While you might think the inner and outer curves of the arch look like parabolas, they are actually a similar-looking shape called a catenary curve. This is the shape assumed by a hanging chain or rope that is suspended at both ends (which is the source of its name, from the Latin for chain). This drooping shape allows the rope or chain to have the least possible potential energy as it hangs. Inverted to form an arch, it provides the strongest shape for the arch.

Mathematicians writing equations for a catenary curve use hyperbolic cosine functions. Trust me, you don't want me to try and explain that here, just take my word for it. If the horizontal ends of a hanging chain are sufficiently close together, and the chain does not hang too loosely, the difference between a catenary curve and a parabola will not be seen by the naked eye, but it is there. And when the scale gets bigger, the difference is significant. Engineers and architects need to be aware of it, but most people would be happy to just call it a curve.

If you knew this, and didn't have to use elimination to get it, you have one up on me! While it looks vaguely like a soccer ball, due to the fact that it was formed by truncating, or cutting off, the vertices of a larger polyhedron, this one only has 16 faces, not 32. The official description says it has 4 sets of 3 pentagons arranged in a tetrahedral arrangement, with 4 hexagons in the gaps. In English, the three green pentagons you can see have a corresponding set on the other side, as is the case for the yellow ones of which you can see two. You can see three of the four hexagons. It would be clearer if we could have a rotating image, but for now some imagination is needed.

So the word truncated tells us that something had its vertices cut off. In this case, that was a triakis tetrahedron, which can be described as a tetrahedron that has had a triangular pyramid built up on each face. This means the figure has 12 faces, three pointing up from each of the four sides of the tetrahedron. To get the shape shown, we don't cut off all of the vertices, only those where 6 edges meet. Looking at the image, you should be able to imagine the four hexagons as being oriented towards the four vertices of that original tetrahedron, and they are formed in the act of truncation.

What use is this? I cannot give you a real-world example of a truncated triakis tetrahedron in action, so maybe not much. But it is pretty cool! And it offers a taste of the many complex shapes that can be formed, interesting for themselves and their properties, not just for their practical applications.