I read the book in the 90s. I don't have a clear recollection of how old it may have been at the time, but I think it was relatively new. I think that there were a young man and a young woman in the story. The young man was the protagonist. I believe he first encountered the other dimensions through a portal, possibly in his neighbor's basement. I think the neighbor was an elderly man. I remember that he went to a two dimensional world first. I also remember there were words for the direction he had to move to get to the fourth dimension, possibly "kaa" or something similar?

Finally, it may be this same book or it may be a different book I read around the same time, but I think he may also have had some technology he got from his neighbor that allowed him to speed up and slow down time. I believe the control for this device was described as running down his sleeve with a knob in his hand which he turned one direction to speed up and the other to slow down.

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    The cardinal drections in a fourth spatial dimension are commonly known as "ana" and "kata," terms coined by the nineteenth-century SF author Charles Howard Hinton. – Buzz Jan 16 at 6:17
  • The first part reminds me of Flatland. – Raj Jan 16 at 14:11

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