Rotations and Reflections

Congratulations, you solved the ancient Greek paradox of Zeno. Euclid was so happy with your explanation that he decided to help you!

He looked through the code and found a mistake in your tables of rotations and reflections. Your time machine does a series of translations, rotations and reflections through four-dimensional space. Unfortunately, your programmer put in the wrong coordinate transformations in your master table.

Euclid is too excited about building his own computer to help you complete the task so you're on your own. Simply select the name of the transformation that corresponds to each of the possible coordinate changes below. There are four rotations (90, 180, 270 & 360°) and four reflections (across four different lines: the x-axis, the y-axis, the line y = x, and the line y = –x).

Good luck!

(If you're thinking about just guessing, there are 8*7*6*5*4*3*2*1 = 40,320 possible answers!!!)

Keep x & y	New x: keep sign	New x: change sign
New y: keep sign	(x,y) → (x,y) select one... Rotation 90° Rotation 180° Rotation 270° Rotation 360° Reflection x-axis Reflection y-axis Reflection y = x Reflection y = –x	(x,y) → (-x,y) select one... Rotation 90° Rotation 180° Rotation 270° Rotation 360° Reflection x-axis Reflection y-axis Reflection y = x Reflection y = –x
New y: change sign	(x,y) → (x,-y) select one... Rotation 90° Rotation 180° Rotation 270° Rotation 360° Reflection x-axis Reflection y-axis Reflection y = x Reflection y = –x	(x,y) → (-x,-y) select one... Rotation 90° Rotation 180° Rotation 270° Rotation 360° Reflection x-axis Reflection y-axis Reflection y = x Reflection y = –x

Switch x & y	New x: keep sign	New x: change sign
New y: keep sign	(x,y) → (y,x) select one... Rotation 90° Rotation 180° Rotation 270° Rotation 360° Reflection x-axis Reflection y-axis Reflection y = x Reflection y = –x	(x,y) → (-y,x) select one... Rotation 90° Rotation 180° Rotation 270° Rotation 360° Reflection x-axis Reflection y-axis Reflection y = x Reflection y = –x
New y: change sign	(x,y) → (y,-x) select one... Rotation 90° Rotation 180° Rotation 270° Rotation 360° Reflection x-axis Reflection y-axis Reflection y = x Reflection y = –x	(x,y) → (-y,-x) select one... Rotation 90° Rotation 180° Rotation 270° Rotation 360° Reflection x-axis Reflection y-axis Reflection y = x Reflection y = –x

Complete the tables to fix your time machine!
 

Zeno's Paradox

Unfortunately, you took too many guesses to realize the tetrahedron is the logical choice for a primitive solid. Euclid doesn't believe you are really from the future and says to, "Now leave me alone. I've got a puzzle to solve."

Before leaving, you look over his shoulder and see a diagram of a running man where in each time-increment, the runner moves half way to the finish line. You recognize this as Zeno's paradox. It shows that motion is impossible according to the following proof:

1. Suppose a runner always moves half the distance to the finish line during each step.
2. Now allow an infinite number of steps.
3. Since the runner can never reach the finish line even with an infinite number of steps, the runner will never reach the finish line.
4. Therefore, motion is an illusion.

You wonder if you could resolve the paradox and explain it to Euclid, maybe he would be so grateful that he'll help you fix your time machine. It's worth a shot. Now you just have to figure out what's wrong with the proof.

Good luck.