The game is simple: we have a octagon cut out in the cardboard and on each of the Two faces one is designed arrow. The first arrow is inclined to 0 ° compared tovertical axis – we could say that “marks 12” – while the second arrow is inclined to 90 ° compared tovertical axis – In jargon “scores 3”.
Doing Turn the octagon along the vertical axiswhich coincides with the axis on which the first arrow is located, we see the two arrows alternate on the octagon “between 12 and 3”. Now, taking into consideration the face in which the “of 12” arrow is located, let’s do rotate the octagon of 45 ° clockwise, thus bringing the scum on the Second summit of the octagonas in the figure below.
Now let’s go back to performing the rotation along the vertical axis That, in the initial configuration, the two arrows between 12 and 3 alternately appeared to us. Where do you expect the arrow of the face that is shown after the rotation? If you thought at the bottom right … you were wrong. Watch the video!
It looks incredible but the arrow it seems not to move! After rotating the plane containing the octagon, in fact, the two arrows will show themselves in the same position itselfthat is, on the second summit at the top right. But how is it possible?
This geometric trick has to do with the rotation axes. Let’s try to understand why.
In the initial configurationwe positioned one of the two arrows on the central top top, to form 0 ° with the vertical axiswhile the second arrow was designed on the rear face a 90 ° clockwise compared to the vertical axis. When we have wheel of 45 ° the octagon in clockwise Compared to the front face, the rear face instead made a rotation in anti -clockwise! And this means that, while the first face went from 0 ° to 0 °+45 ° (hourly rotation) that is 45 °, the second has gone from 90 ° to 90 ° -45 ° (anti-eral rotation) that is always 45 °!
The result is therefore that both arrows are now inclined by 45 ° compared to the vertical. But why will we think that the second arrow, after the 45 ° rotation, will be in the diagonal summit at the bottom right? The answer lies in the fact that in the first configuration, We rotated the octagon according to the vertical axis which in this case coincided with the axis of the first arrow. In the second configuration, however, we continue to rotate the octagon according to the vertical axisbut no longer coincides with theaxis relating to the first arrowbut our brain continues to think according to its position!
In fact, if we do turn the octagon according to the 45 ° inclined axis compared to the verticalthat is, the axis of the first arrow, we see that the rear arrow will position itself right where we imagined, that is at the bottom right.
