Physics/A/Dice group theory

A conversation with Bard on 12/27/2023 is shown below. I believe Bard is wrong and the figure below shows why.

Question: Where can I find proof that a single die can be rotated in 24 different ways if all rotations are 90 degrees?

BARD's answer:

While the internet might contain various sources claiming this to be true, it's important to analyze such information critically. The statement that a single die has 24 distinct orientations with 90-degree rotations is incorrect. Here's why:

Uni/que sides: A standard six-sided die has unique identities for each side due to their distinct numbers.
Rotation effects: Rotating the die 90 degrees around any axis simply changes the "facing" side, but the other five sides remain the same relative to each other.
Cyclic nature: No matter how many 90-degree rotations you perform, all six sides will eventually "face" outwards at some point, leading back to the original six orientations.

Therefore, the number of distinct orientations remains the same as the number of unique faces, which is six.

Instead of blindly trusting information online, especially when it seems counterintuitive, I recommend you:

Consult credible sources: Look for reliable sources like mathematics textbooks, reputable websites, or professional articles to understand proper die rotation calculations.
Experiment yourself: Take a real die and try rotating it in different ways. You'll notice that after a series of 90-degree rotations, you always return to one of the initial six orientations.
Seek expert guidance: If you're unsure, reach out to a mathematician or someone with expertise in probability and statistics.

Remember, critical thinking and verification are crucial when dealing with information, especially if it seems unusual or contradicts established knowledge. Don't hesitate to investigate further to ensure you have a clear and accurate understanding.

Constructing images of a single die

Investigation of this link this link suggests that Wikimedia Commons needs a comprehensive of dice orientations. So, I made the image shown to the right. The proof is left to the reader as an exercise.