I can't recognise the 八 radical in many characters I have my oxford and tuttle dictionary in front of me and the radical list has a lot of characters with 八 as it's radical, which means "divide" apparently.

But where is this radical in 兰 ，半 ，弟， 单 etc ?

Is it written in mirror in those characters? Thank you
丷 is a variant of 八, just as 灬 is a variant of 火.

Variant characters are common for radicals, given that they can occur at difference positions within a composite character.