Talk:Dunce hat (topology)

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How to make one

Take a circular sheet of paper with centre O and draw three radii OA, OB and OC with angles 120° between them. Fold along the radius OB and crease, at right angles, along OA and OC bringing those two radii together. Use plastic tape to fasten the arcs BA and BC together. The arc AC should now be the base circle of a cone with apex O. Fold along OC completely so that the arc BC wraps round the circle taking B to A=C and tape that arc into place too. It is then a model of Zeeman's dunce hat.

It is not a manifold because points on the base circle do not have a neighbourhood that is a disc. It is a cell complex consisting of a 1-cell attached to a 0-cell making a circle to which a 2-cell is attached. It is trivially CW. I don't have access to Munkres book but none of the spaces defined by Krarinettus are homeomorphic to the Dunce Hat. A generalisation would be to divide the circular boundary of a disc into 2n + 1 equal arcs and glue them with n in one direction and the other n + 1 in the other.

There is a simple way of contracting it (though not the paper model). The point A can be pulled up to a point, say A', on the radius OA by squeezing two cones of points near A in through the point A so that they then occupy the segment AA'. However, the contents of the segment that come from the two sides of A are only connected at A' and held together but not connected at A. They can be pulled apart leaving A and A' fixed. Since there is now a hole in the disc, it can be stretched out to clear the disc's interior. All the points will then be in the circular base which is not connected at A and can be contracted to one of its points. Hubert Shutrick — Preceding undated comment added 22:30, 9 December 2014 (UTC)


As far as I can see, the dunce cap has a subset that looks locally like THREE 2-cells joined by a 1-cell, so i don't think this is a manifold. If you only loosely know what "manifold" means, then this might make more sense than something you don't understand at all like, say, algebraic variety, or CW-complex, or stratified manifold, (all of which can describe this kind of space) but nonetheless it should be changed. Right? MotherFunctor 06:20, 27 April 2006 (UTC)


the gif is very confusing. after reading this article, i still have no idea what this is. (talk) 23:54, 5 February 2009 (UTC)

The gif starts from the flat triangle, with a hole cut out of the middle (blue ring). It then tries to glue up the sides of the triangle as required. This can't be done within 3-space without self-intersections, so the triangle gets folded through itself to allow the green edge to come together in the right way. Jebanks (talk) 15:37, 8 July 2014 (UTC)


I believe "Simply gluing two sides oriented in the same direction would yield a cone much like the layman's dunce cap" should read "...oriented in the OPPOSITE direction..." because gluing two sides with the same orientation and leaving the third untouched (fundamental polygon symbol = aab) would yield a projective plane with an open disc removed (Möbius Strip), whereas it is easy to see visually that gluing two sides with the opposite orientation and leaving the third untouched (fundamental polygon symbol = a(a^-1)b) yields the desired cone. See Munkres Topology 2nd Edition pp. 448.

Furhermore, Munkres pp. 443 gives the definition of the n-fold dunce cap as the quotient space of the unit 2-ball under the equivalence relation defined by identifying each point of the boundary (circle) with those points that are rotations of the original point by an integer multiple of 2pi/n radians. Thus Munkres' definition is equivalent to an oriented n-gon with all sides identified and having the same orientation. For n=2, this yields the projective plane, and for n=3 (the case discussed in this article) no sides should have their orientations reversed, meaning both the definition and image to the right are incorrect.

All that being said, I am only a student taking Topology for the first time so my understanding is hardly authoritative -- can someone take a look and sort this all out? Klarinettus (talk) 22:24, 30 April 2009 (UTC)

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