ju.hrozian/Shutterstock
Why are at least four faces required to make a flat-surfaced solid in 3D? And does a similar lower limit apply in higher dimensions?
Mel Earp
Macclesfield, Cheshire, UK
When dealing with higher dimensions in geometry, it is often instructive to start with lower ones and work your way up.
If we start with two points, which are by definition each of zero dimension, we can join them together to form a line, which is one-dimensional. Two points is the minimum. In order to progress into two dimensions, we have to add another point not on the line to make a triangle, which we can then fill in to make it “solid”. This is the minimum straight-edged, plane figure in two dimensions, and its boundary is formed from three one-dimensional edges. So, in two dimensions, there must be at least three lines on the boundary.
To get three dimensions, we have to add another point – not on the same plane as the triangle – and we get a tetrahedron. This is the minimum number of points. This figure has four plane, triangular faces making up its boundary and it is the minimum number possible because we couldn’t get there any simpler way. These triangles are formed by taking a combination of any three of the four vertices.
If we now continue up the dimensions, we have to add another point at each step, taking care to move into the new dimension just as we did in the two and three-dimensional cases. Then we can see that at dimension n we need n+1 points to make the vertices of this figure, which is called an n-simplex. The boundary of this figure is made up of n objects of dimension n-1, known as (n-1)-simplexes, formed by taking all the combinations of n of the n+1 vertices.
So, as an example, a 4-simplex in four dimensions is formed from five vertices and has five tetrahedra (which are 3-simplexes) as its boundary. It is also possible to calculate how many lower-dimensional simplexes an n-simplex contains.
This means that a tetrahedron has four triangular faces, six straight edges and four vertices, which are formed by taking combinations of three, two and one vertex, respectively.
In general, the number of r-simplexes that an n-simplex contains is the number of combinations of selecting r of the n vertices. This is (n!/r!(n-r)!) where “!” is the factorial operator.
Ron Dippold
San Diego, California, US
The answer to the lower limit part of the question is yes. And to see why, it is very helpful to take a look at lower dimensions. Take a straight line, a one-dimensional space (1D). How many zero-dimensional faces (points) does it take to demarcate an object on the line with any length? Two points (1D + 1).
Now look at a flat plane, a two-dimensional (2D) space. How many flat, 1D faces (straight lines) does it take to outline a full 2D object? It takes three (2D + 1), a triangle. As the questioner noted, in a three-dimensional space it takes four (3D + 1) flat surfaces to enclose a 3D object (a tetrahedron, also called a triangular pyramid). If you go up to four dimensions (4D), an object called a 5-cell or pentachoron is a 4D “triangle” bounded by five (4D + 1) tetrahedra. And so on.
Why n+1? Go back to the 2D triangle. If the questioner hadn’t specified flat faces (straight lines here), we could do it with one line – a circle. Using straight lines, this takes three lines because you have to enclose a non-zero 2D area such that it is completely separated from the outside. You can see this easily with a 2D triangle – there is no way to enclose an area with only two straight lines.
Now imagine this on a grid. If we stick to the major dimensions, a laser inside can shine four ways: north, south, east or west. This is always the number of dimensions times 2 because the laser can shine either way (N or S, E or W) along each major dimension. So if you stick to lines on the grid, you need four lines (two times the dimensions) to make a rectangle that traps all the light. In 3D, it is six faces for a box. But if you “cheat” and don’t stick to the grid, each line can block lasers in two directions.
“
When dealing with higher dimensions in geometry, it is often instructive to start with lower ones and work your way up
“
If you have a slightly rotated triangle, the first line can block lasers going W and N (from inside), the second line can block lasers going N and E, and the third line can block lasers going S and W. So each line blocks two directions of laser, but in every case (try it yourself) there is one direction left over, S here, that needs an extra line to block it. In the end, you need (directions/2 )+1 lines, which is dimensions + 1. This holds in higher dimensions too – each 2D face blocks lasers from inside in three directions. Try it yourself with a tetrahedron!
John Elliott
Bramhall, Greater Manchester, UK
In 1D, two points are needed to define a straight line. In 2D, three straight lines are needed to define an area. In 3D, four such areas are required to define a volume. One may suppose this pattern applies in any higher n-dimensional space, too. But for us humble beings confined in 3D, it is hard to visualise beyond that.
To answer this question – or ask a new one – email lastword@newscientist.com.
Questions should be scientific enquiries about everyday phenomena, and both questions and answers should be concise. We reserve the right to edit items for clarity and style. Please include a postal address, daytime telephone number and email address.
New Scientist Ltd retains total editorial control over the published content and reserves all rights to reuse question and answer material that has been submitted by readers in any medium or in any format.
