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| - | '''Digit notation''' uses a single number to represent a [[hypersphere]], and combinations of the numbers to represent the [[cartesian product]]s of those shapes. Thus, '1' is a [[line (object)|line]], '2' is a [[circle]], and '3' is a [[sphere]]. A shape with only 1s in its sequence of digits is a [[hypercube]]. Thus, '1' is a line, '11' is a [[square]], and '111' is a [[cube]]. The sum of the numbers in a shape is the dimensionality of the shape.
| + | #redirect [[Rotope#Notations]] |
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| - | The digits in any particular sequence are interchangeable, so a sequence of digits refers to the same shape when any of its digits are transposed. For example, ''21'' and ''12'' both represent a [[cylinder]]. Another result of this invariance under transposition is that the same shape can result from cross products of different shapes. Thus, the sequences ''1212'' and ''1122'' both represent the same shape, which can be constructed from any of the following cartesian products - the product of two cylinders, the product of a square and a [[duocylinder]], or the product of a [[cubinder]] and a circle, or any other combinations of its subshapes. If a sequence contains a digit greater than 1, it is capable of rolling. If all of a sequence's digits are greater than one, then it will always roll when placed on a surface. If it only has 1s then it only has flat sides and is incapable of rolling. All of these shapes can be refered to by their signatures.
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| - | Two extensions were later added to the digit notation. These are [[taper]]ing and [[spherate|spheration]]. A superscript number 1 means that the shape formed so far will now be tapered to a [[point (object)|point]]. A superscript number greater than 1 means that it will be tapered that many times. A pair of parentheses surrounding a group of digits represents that that group of digits will be spherated. Groups seperated by parentheses are independent from anything outside those parentheses; i.e. ''1(21)'' means that only the ''21'' part is spherated and not the first ''1'' (though this particular shape is usually written as ''(21)1'').
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| - | == Conversions ==
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| - | === To group notation ===
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| - | #Where there is a non-superscript digit ''n'' greater than 1, change the digit to a sequence of ''n'' 1s, and surround them with parentheses. | + | |
| - | #Where there is a superscript digit ''n'' greater than 1, change the digit to a sequence of ''n'' 1s, keeping the sequence in superscript, but do not surround them with parentheses.
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| - | #Replace every digit 1 with a letter.
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| - | === To product notation ===
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| - | [[Marek14]] and [[bo198214]] found [http://teamikaria.com/hddb/forum/viewtopic.php?p=9169#p9169 a way to convert digit notation to product notation]: | + | |
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| - | #Parentheses evaluate from inside out.
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| - | #Parentheses of for<span>m</span> a1111... with b 1's evaluate to a#(b+1)
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| - | #Other parentheses containing any 1's evaluate to a#(b+k) where a#b is evaluation of the same parentheses without 1's, and k is the number of 1's.
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| - | #Parentheses containing terms a,b,c etc. with at least two terms and none of them equal to 1 evaluate to (a x b x c x ...)#n where n is the number of terms.
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| - | [[Category:Geometric notation]]
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