chalkdust we can also make statements about all possible elections run under a voting system, such as “the alternative vote is clone-proof”. Many diﬀerent voting systems are in use around the world, and it is a mathematical certainty that every electoral system lacks some desirable characteristics, so when discussing which system is the “best” it can only ever be a debate about which criteria people think are most important, and that may depend heavily on the context of the election.

Alexander Bolton is a PhD student at Imperial College London, applying Bayesian change point models to detect regimes in multivariate stochastic processes.

References and further reading Arrow KJ (1950). A diﬀiculty in the concept of social welfare. Journal of Political Economy 58 (4) 328–346. Johnson PE (2005). Voting Systems. hp://pj.freefaculty.org/Ukraine/PJ3_VotingSystemsEssay.pdf May KO (1952). A set of independent necessary and suﬀicient conditions for simple majority decision. Econometrica 20, 680–684. Woodall DR (1994). Properties of preferential election rules. Voting maers 3, 8–15.

My favourite shape

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 ploing 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! chalkdustmagazine.com

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