Error coins are not to be confused with coin die varieties.
A double die error is a production error caused when the initial die was created from a master die or hubbing die. The double die occurred in the production of the die not the coin.
An error coin is a coin which contains a production defect which becomes part of the coin.
Coins can be struck more than once or even twice.
If the coin blank is able to move you will see double or triple images.
Proof coins were purposefully double struck in order to get very high relief but the coin was fixed to prevent movement.
Errors can be from an impression of rag fibre left on the die while cleaning it.
They can be a miss strike where the planchet (coin blank) is miss fed into the coin press and only part of the coin design is struck into the surface of the coin.These are called off centre and usually given a percentage like 10%, 30% or even 70% off centre.
In other cases you can have delaminating or separation of the various metals which make up the coin.
I had a coin that looked like a pie with piece neatly cut out extending only about half the thickness of the coin and the coin design was still trying to extend into this void in the coins surface.
Other more interesting errors are when a coin blank which was struck fails to eject from the coin press and it then comes down in contact with another blank but this time the reverse impression is pressed into the new coin blank below causing one side of the coin to be struck with it relief out and the other side with the relief the same but in.
This creates the two headed or two tails coin with one side a positive image transfer and the other a negative transfer not from the die but the coin stuck in the press acting like a die.There are also various rim problems which are collectable when the edging device fails and only part of the coin receives its edge surfacing.
There are people that collect as many examples of errors that they can find.
Are they rare? Not really. Are they collectable, certainly? Valuable? Not really. Interesting? Certainly!
The most valuable error coins are those in which dies are improperly paired. Such as a cent obverse and say dimes reverse or when a design change occurs and both the old and new designed dies are used on the same coin.
I find collecting errors of coins produced in old roman times particularly interesting especially since they were hand struck.
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