In the context of hypercubing, a physical puzzle refers to an N+1 dimensional puzzle projected in N dimensions, such that the projection is operationally equivalent, i.e. the projection "emulates" the true puzzle. Usually, if not specified, it is used to refer to a 4D puzzle projected into 3-space.
2D Physical Puzzles
When designing 3D physical puzzles, it's a good idea to try and step down the problem by first looking at what the 2D physical analog would look like. For example, the 2^3, or by it's more recognizable name, the 2x2x2, can be projected down into 2D by splitting it in 2 halves and putting them next to each other along the X or Y axis. By doing this, the Z axis will coincide with the X or the Y axis, which will make certain moves inaccessible without the use of a gyro algorithm to reorient the puzzle (the gyro algorithm does a cube rotation). We can also see that the projected cubies aren't square shaped, and that is because a square doesn't have 3 fold symmetry.
By the same logic, we can then build a 3D physical 2^4/2x2x2x2 - split the hypercube in 2 halves and put them next to each other along X, Y or Z axis, making certain moves inaccessible without the use of a gyro. We can see that the 2^4 can have the cubies be cube shaped because (by mathematical coincidence) a cube has 4 fold symmetry (a 2^4 cubie has 4 colors).
Another thing to note is that physical puzzles don't have a true mechanism/way to hold the pieces together, so magnets are used to hold the puzzle together. Because we don't have a true mechanism, we need to limit the moves we are allowed to do by the canonical moves to make sure we aren't doing illegal moves on the puzzle.
Another example is the 3x3 pyraminx. In this case we can take a top-down orthographic view of the puzzle and then project it, making sure each piece type has the same shape in the projection so they are interchangeable. Doing this, we get an interesting 2D physical puzzle that looks like the wireframe of a tetrahedron projected in 2D, but with some of the lines disconnected. This is because we can't project a tetrahedron into 2D without unevenly distorting the shape or without having breaks in the shape. This can then be scrambled and solved using its canonical moves (rotate 3C pieces with its 2Cs, by first making sure all its associated 2Cs are touching it). By the same logic one can construct a 3D physical simplex.
3D Physical Puzzles
The first physical puzzle was the 2^4, designed by Melinda Green in 2017. After this Grant, Luna, Hactar and Markk designed and built many other 3D physical puzzles based on Melinda's 2^4 design, like the 3^4, the simplex, and hypercuboids like the 2x2x3x3. There where attempts at designing shapeshifting physical puzzles with no success.
3D Physical designs list
Below is a table of physical puzzles, both produced and unproduced.
|Puzzle||Name(s)||Date Design Finished||Date Construction Finished|
|2x2x2x3||Luna & Grant||6th December 2021||3rd February 2022|
|2x2x3x3||Grant & Hactar||17th January 2022||14th May 2022|
|2x3x3x3||Grant||17th January 2022||6th July 2022|
|3x3x3x3||Grant||8th February 2022||22nd July 2022|
|1xAxBxC series||Grant||3rd May 2022||Not constructed|
|1x1xAxB series||Grant||12th May 2022||Not constructed|
|Simplex||Markk||30th August 2022||Not constructed|
|pretty much any cuboid||Grant|
Physical Shapeshifting Puzzles
Physical shapeshifting puzzles are hard to design, if not impossible, because of their shapeshifting nature. All the current 2D and 3D designs don't fully work, in some solved states appearing as they're scrambled.