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Anybody know of "serious" mathematical ornaments or toys like the Gömböc, etc?

Already have a rubix and abacus (that's more of a tool though).

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    You probably mean "lax" in the above comment (somewhat Ironic since in your question statement it probably should be "Rubik's" cube).2012-01-18

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In a blatant reference to my own creative work, I submit for your consideration the mathematical artwork presented on my Shapeways Shop at this address: https://www.shapeways.com/shops/Feingold_Math_Art But there are a huge number of other artists whose work can also be found in other shops there.

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For when the Möbius strip is too pedestrian, the good people at Acme Co. claim their Klein bottles are the "finest closed, non-orientable, boundary-free manifolds sold anywhere in our three spatial dimensions."

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    His book the Cuckoo's Egg is a lot of fun, too.2012-01-18
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On Shapeways, you can find a variety of mathematical ornaments.

My favorites are the differential geometric surfaces designed by Bachman. I also like Bathsheba's designs as well.

enter image description here

(Sorry the image is so large. Is there a way to reduce the size?)

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I have a Rattleback at my desk. Fun to fiddle with while thinking.

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The Rubik cube. ${}{}{}{}{}{}{}$

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    Oh yeah obviously I have one of those - I have some with more than 6 faces also, though in the end it's a tougher algorithm. Very nice puzzles though.2012-01-18
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Spirograph from Hasbro. You can make lots of famous mathematical curves with its pieces: epicycloids, hypocycloids, etc. MathWorld has an article on some of these curves.

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    I love the Spirograph; +1 of course.2012-01-26
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Zometool is a construction kit which has 2-, 3-, and 5-fold symmetry, which is great for building (3D projections of) the 120-cell, or just for playing around.

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    @Adam: Seeing this, I thought of K'nex. Their website doesn't have any info on what symmetries are supported on the home page. I grew up with tinker toys.2012-01-18
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Wikipedia has a section on mechanical puzzle.

As for myself, my fascination with algebraic topology began with metal link puzzles or hanayama.

Of course, Tower of Hanoi is a classic.

Of interest may be the reference: Adventures in Group Theory: Rubik’s Cube, Merlin’s Machine, and Other Mathematical Toys by David Joyner. Book description in Amazon.

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While it may be a 'children's toy', the Switch Pitch works remarkably well as an object of mathematical sculpture; it's fundamentally based on the fact that the vertices of a regular cube are also the vertices of two (interlocked) regular tetrahedra (if your cube is $\{0,1\}^3$, take the vertices with $i+j+k$ respectively odd or even). It helps that people can't help but fiddle with it; it's been a perfect hand-fidgetting toy for me.

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The Oloid is fun to touch, to watch and understand.

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Maybe a Tippe Top.

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Love zometool [1]

[1]: http://www.zometool.com/ for geometry, platonic and archimedean solids, among others/

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Double pendulum.

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http://tavernpuzzles.stores.yahoo.net/puzzle.html http://ep.yimg.com/ca/I/tavernpuzzles_2188_4561747

Goal: Remove the big ring

http://tavernpuzzles.stores.yahoo.net/puzzle.html

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I think there are 2 different toy devices that mimic a Gray code. One can be seen at http://mypuzzlecollection.blogspot.com/2011/12/brain.html. The other is named "Spin Out".

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    Also the Towers of Hanoi.2012-05-14
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Sudoku and all other logic trainers would also fit.

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Check Montessori mathematical materials like the binomial cube just google it

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Origami to fold shapes to find surface area of 3 dimensional shapes. Wooden blocks. The Game of 24. Math Jeopardy. Bucky balls. Sudoku. Computer games with rotations and other types of transformations.