The word 'order' is a noun form as a word for:
The noun 'order' is a word for a thing.
The noun forms of the verb to order are orderer and the gerund, ordering.
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The word 'arrangement' is a noun, a singular, common, abstract noun; a word for the order in which things are put, or preparation or planning done in advance.The word 'arrangement' is the noun form of the verb to arrange.
The noun form of correction might be the word correction. A correction is defined as a change that is made in order to create a specific result. When describing this change, the word correction would be a noun.
The noun 'discipline' is an abstract noun because it is a word for a concept. The noun discipline is a word for a field of study (a subject); a system of rules governing conduct; control gained by enforcing obedience or order; the practice of training by use of punishment.
Equilateral is an adjective, not a noun and so the question makes no sense. The order of rotation of an equilateral quadrilateral, such as a rhombus, is 1.
The word 'apprentice' is both a noun and a verb. The noun 'apprentice' is a word for someone who works at a job in order to learn a skill, a word for a person. The verb 'apprentice' is to employ someone to train in a skilled job, or to work for the purpose of training in a skilled job.