Top ten vote issue 04

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Page 3 model: Hallucinations

Hallucination imageYou might think that maths and psychedelic hallucinations tend not to mix very well. But you would be mistaken! There are a series of visual hallucinations known as form constants that are highly geometric, and a mathematical model of them has provided us with some fascinating insight into how our visual cortex (the part of the brain that processes the information we receive from our eyes) works.

These hallucinations were first observed in patients who had taken mescaline, a psychedelic drug produced from a cactus found in South America. Form constants have subsequently been reported in a number of other altered states such as sensory deprivation, waking/falling asleep states, near death experiences and by individuals with synaesthesia. Some people even report seeing these patterns after closing their eyes and applying firm pressure to both eyelids for a few seconds!

Hallucination imageThe mathematical model we referred to was described in a paper by Bressloff et al., and is based on anatomical features of our brain. It seems that the visual cortex has certain symmetry properties, such as reflective, translational and even a novel shift-twist symmetry. Its electrical activity can be represented mathematically and—a bit of group theory, some eigenvectors and a couple of transformations later—has steady state solutions to the resulting equations that are remarkably similar to the observed hallucinogenic experiences. Groovy!

Disclaimer: Chalkdust does not advocate pressing hard on your eyelids.

[Written in collaboration with Samuel Mills. Pikachu adapted from picture by Matt Levya, CC BY 2.0; Hallucination pictures taken with kind permission from PC Bressloff, JD Cowan, M Golubitsky, PJ Thomas and MC Wiener, What geometric visual hallucinations tell us about the visual cortex, Neural Computation 14(3) (2002), 473–91.]


Prize crossnumber, Issue 04

Our original prize crossnumber is featured on pages 48 and 49 of Issue 04.


  • Although many of the clues have multiple answers, there is only one solution to the completed crossnumber. As usual, no numbers begin with 0. Use of Python, OEIS, Wikipedia, etc. is advised for some of the clues.
  • One randomly selected correct answer will win a £100 Maths Gear goody bag. Three randomly selected runners up will win a Chalkdust t-shirt. The prizes have been provided by Maths Gear, a website that sells nerdy things worldwide, with free UK shipping. Find out more at
  • To enter, submit the sum of the across clues via this form by 7 January 2017. Only one entry per person will be accepted. Winners will be notified by email and announced on our blog by 21 January 2017.

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Dear Dirichlet, Issue 04

Dear Dirichlet

Moonlighting agony uncle Professor Dirichlet answers your personal problems. Want the Prof’s help? Send your problems to

Dear Dirichlet,

This year has been a bit tough for us financially, and the only summer holiday we were able to afford was a week camping in the not-so-sunny East Midlands. My work colleagues, however, won’t stop talking about their blissful trips to the beaches of southern Europe. I don’t want to seem jealous but I wish they’d stop chatting about their suntans. How can I move the conversation on?

— Beyond the pale, Selly Oak

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