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Is there any particular reason why the pips (the little dots) on a die are arranged in those patterns (3 is a diagonal line, 6 is two horizontal lines, etc.)? And also why any face plus its opposite equals 7?

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This is a really interesting question and just spent last hour googling around for various thoughts.

The first thing is why pips and not numbers. This is because the invention of dice predates the invention of numbers. source

According to the wikipedia article on pips its notes that the pip designs are 'easily countable' Why the particular patterns for the numbers I don't know the fact is if you roll a 5 you recognize the pattern straight away. Its worth noting the the pip design on a six sided dice is the same pip pattern on standard playing cards for values Ace-6. Someone may be able to explain the origin but I imagine its a similar answer to why is are Letters shaped they way they are? I's assumes the pip pattern is just something that's evolved over time that we then recognise. if a dice (or a playing card for that matter) just displayed pips in a long row then counting them each time would be very tricky so the fact they evolved into a pattern that we regonise without counting the pips is probably where it came from.

Finally, why do numbers on opposite sides add up to the same number? According to to this article Dice were not always like that, apparently some Roman dice were arrange with opposite sides adding up to prime numbers. ie 1 opposite 2, 3 opposite 4 and 5 opposite 6. This article states that if a dice is random it really doesn't matter what sides the numbers are on, However the article suggests the rise in opposite sides adding up to 7 appears to be that was because it is perceived to be more fair, even though its still random what number comes up. From the article:-

“We think users of dice also adopted new ideas about fairness, and chance or probability in games,” said Eerkens in a statement. The numbering style changed from the prime number configuration (1-2; 3-4; 5-6), one heavily influenced by popular ancient Egyptian numerical arrangements, back to the style “where opposite sides add up to seven (6-1; 5-2; 3-4).” Eerkens suggests that while the direct cause for this shift is unclear, it could possibly be related to an gradual effort to make fair and balanced dice.

Also I should say that someone else with more historical or archaeological knowledge than I might be able to give a different answer. I'm really intrigued by the idea that opposite numbers on a dice adding to same number is perceived to to be more fair that any other arrangement despite it not making any difference.

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    One minor correction: the pip design on a six sided dice is the same pip pattern on standard playing cards for values Ace-6. Not exactly, the "3" on a die form a diagonal line, while the three symbols on a "3"-card are in a vertical line. – Volker Landgraf Nov 30 '19 at 16:14
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    I had also noticed that the pips on dominos 1-6 use the same patterns as dice. Thank you for the detailed answer! – user45266 Nov 30 '19 at 18:34
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Rotation Invariance

The reason the pips are configured in the pattern they are is so they look the same no matter what angle you're looking at them, as much as possible. This is called rotation invariance.

There are differences in the Occidental style and the Oriental style of dice; with the 2-side, for instance, the Occidental style has them at opposite corners, and the Oriental style has them centered. These are both ways to achieve the same purpose.

http://www.aleakybos.ch/config.htm

Numerical Balancing

The reason for having opposite sides add up to 7 is for numerical balance.

In a die that was manufactured perfectly, this may not matter, but in fact most dice have small manufacturing defects like bubbles, or being tumbled more on one side than the other, so some sides are heavier than others.

By configuring the die with numerical balance, you can compensate for some of these manufacturing defects. In a die where all the high numbers are clustered together on one side and all the low numbers are clustered on the other, having one side of the die heavier than the other would make it roll unfairly.

In ancient and classical times, numerical balancing wasn't seen as important, because a die roll was seen more as controlled by gods or the fate than by what we know as random chance.

During the Renaissance, dice underwent yet another significant change. Starting around 1450, they became less regular in size and pip style, but more standardized in symmetry and configuration, which shifted back to the “sevens” system. The increasing attention paid to symmetry in particular may have been driven by new knowledge of probability, a field of mathematics that blossomed during the Renaissance.

https://www.smithsonianmag.com/smart-news/how-centuries-old-dice-reveal-changing-attitudes-about-fate-1-180968090/

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