Prime Factorization Naming Scheme spreadsheet

Kodegadulo
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Kodegadulo
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Joined: Sep 10 2011, 11:27 PM

Aug 24 2018, 10:50 PM #1

[d] All numbers decimal

I have put together a spreadsheet in the Cloud, in Google Sheets, at this link: Prime Factorization Naming Scheme. I'm working it up as an aid in exploring the notion of constructing names for whole numbers based on their prime factorizations. The spreadsheet is visible to anyone with the link to it, but not editable. If you would like to experiment with your own ideas, just make a copy of the sheet.  You should automatically get a space in Google Drive if you don't already have one.

In part, my spreadsheet was inspired by the names icarus came up with for his argam numerals (well known to members of this forum), which embody aspects of this notion. However, argam names exhibit some more artful aspects that add to their attractiveness but which make their construction more complex, and therefore harder to automate (such as with a spreadsheet). What I'm aiming for here is something along the same vein, but much more regularized.

This is also more immediately inspired by newcomer Craig's recent thread A Simple and Intuitive Name and Numeral System for Sexagesimal. Even though Craig's goal was limited to providing digits for a positional base no larger than base 60d, his "datraka" naming convention does embed a similar focus on prime factorizations, with prime numbers getting more or less monosyllabic roots that are agglutinated to form the names of composite numbers.  Craig's scheme is limited to giving roots only to the first three primes (2, 3, 5), but of course argam assigns roots for many, many more primes. But I was intrigued by the consonant-vowel (CV) pattern Craig used, where the root for a given prime starts with a given consonant (or consonant cluster), but then the vowel indicates the power to which the prime is raised.

I've been playing around with this kind of CV pattern for a few days, but ultimately I've found it too limiting.  Not enough room to cover a large number of primes in a reasonably intuitive way.  So what I've opted for is something closer to argam:
  1. I've given each prime a monosyllabic root name with a CVC pattern, that remains stable for all of its powers. So for example 2 = bin, 3 = trin, 5 = kin, 7 = sep, 11 = lev, 13 = yon, 17 = zot, 19 = wox, etc.
  2. I've generally copied icarus's notion of extracting syllables from the name of chemical elements to use as names for corresponding primes. However, I've shaved off trailing vowels or added leading consonants where necessary (usually w-) in order to adhere to the CVC pattern. Hence, zot rather than zote, wox rather than ax, etc.
  3. For raising primes to a power, I devised a common set of suffixes with a -VC pattern, where the consonant is a common thematic element, but the vowel indicates the power.  For example ^1 = -im (usually omitted), ^2 = -em, ^3 = -am, ^4 = -om, ^5 = -um.  Note that rather than ordering the vowel sounds alphabetically, I've ordered them by articulation point, from most open and fronted, to most back and rounded.  My impression is that a fronted sound like /i/, with its generally higher frequency pattern, implies "smaller", while a more back sound like /u/, with its generally lower frequency pattern, implies "larger".  
  4. To get to higher powers than five, I just introduce another thematic consonant and repeat the pattern into another senary place-value: ^6 = -ix, ^12 = -ex, ^18 = -ax, ^24 = -ox, ^30 = -ux.  Combine suffixes for multiple digit powers: -ix = ^6, -ixim = ^7, -ixem = ^8, -ixam = ^9, -ixom = ^10, -ixum = ^11, -ex = ^12, etc.
  5. So powers of a prime are formed by appending these suffixes, e.g.: 2 = bin, 4 = binem, 8 = binam, 16 = binom, 32 = binum, 64 = binix, 128 = binixim, 256 = binixem, etc. I've been generally placing the stress accent on the final syllable in such complexes, so that the suffixes won't all degenerate into schwa sounds.
  6. Composite numbers are formed by juxtaposing the roots (+ suffixes) of its prime factorization. So for example 6 = trinbin, 10 = kinbin, 12 = trinbinem, 14 = sepbin, 15 = kintrin, 18 = trinembin, 20 = kinbinem, 21 = septrin, 22 = levbin, 24 = trinbinam, 26 = yonbin, 28 = sepbinem, 30 = kintrinbin, 34 = zotbin, 35 = sepkin, 36 = trinembinem, etc.  The final constant/cluster of one prime factor is immediately adjacent to the initial consonant/cluster of the next, so this makes the syllable boundaries between prime factors clear enough.
  7. I've opted to order the roots in a composite from largest to smallest. I found that if I did it the opposite way, every other whole number would start with bin, which gets boring; whereas, this way you get more variety up front, yet many numerals wind up rhyming at the end. One could argue that it makes sense to lead with the most significant prime. However, in principle, the order doesn't matter, because multiplication is commutative: 6 = trinbin = bintrin, 12 = trinbinem = binemtrin, etc.
  8. Compare argam's 2520 = kinsevoctove, vs. my equivalent sepkintrinembinam.  Icarus orders from smallest to largest factor, bearing in mind that higher powers of even small primes like 2 and 3 can become larger factors.  Whereas here I am ordering from largest down to smallest prime, regardless of power.
  9. The primorials Pn are: P1 = 2 = bin, P2 = 6 = trinbin, P3 = 30 = kintrinbin, P4 = 210 = sepkintrinbin, P5 = 2310 = levsepkintrinbin, P6 = 30030 = yonlevsepkintrinbin, etc.
  10. I've built the formulas in the spreadsheet so that once you set the name for a given prime, all its occurrences in powers and multiples automatically copy that. Likewise with the list of power suffixes (scroll to the right on the page to find columns defining those). So the spreadsheet can be used to experiment with various choices for syllables.
  11. After exhausting all the currently-known chemical elements (up to atomic number 118, oganesson → wog = 647), I went to Presidents of the United States, from George Washington → wash = 653 to Donald Trump → don = 967 (since I already had Harry S. Truman → trum = 877). Then I went to Vice Presidents of the United States (excepting those that were ever President), from Aaron Burr → bur = 971 to Michael Pence → pents = 1193. And I even dug into the Speakers of the House of Representatives of the United States, from Frederick Muhlenberger → muhl = 1201 to Paul Ryan → ryan = 1571, just 3 primes shy of 1600. (Appropriate enough, since the address of the White House is 1600 Pennsylvania Avenue.) 😉 I'd like to carry this out at least as far as P5 = 2310, so any suggestions for finding more sources of names would be welcome.  (Hmm, maybe Supreme Court Justices?  Senate majority leaders?  British Prime Ministers? State names? State capitals? ...)
  12. This is a naming scheme only. I'm not accompanying it with any notion of how to draw numeral symbols based on their prime factorizations.  I'd defer to something like UTL Numerals, which was the best scheme I've ever seen for doing that. Too bad the anonymous author only posted the images on Facebook for a brief period, until they were taken down suddenly. To my knowledge they have never resurfaced anywhere else.  So all we have left is the image I put together of the first block of 210 (P4) numerals. Whoever this person was, they had filled up the first catalog of 2310 (P5) numerals and looked poised to go on to extend that to 30030 (P6) at least. It was truly remarkable. If anyone knows who the author was, or where UTL has gone now, please let us all know!
So the first block of names is:
Decimal
Dozenal
Numeral Name
1
1
un
2
2
bin
3
3
trin
4
4
binem
5
5
kin
6
6
trinbin
7
7
sep
8
8
binam
9
9
trinem
10
A
kinbin
11
B
lev
12
10
trinbinem
13
11
yon
14
12
sepbin
15
13
kintrin
16
14
binom
17
15
zot
18
16
trinembin
19
17
wox
20
18
kinbinem
21
19
septrin
22
1A
levbin
23
1B
fluor
24
20
trinbinam
25
21
kinem
26
22
yonbin
27
23
trinam
28
24
sepbinem
29
25
nev
30
26
kintrinbin
31
27
sod
32
28
binum
33
29
levtrin
34
2A
zotbin
35
2B
sepkin
36
30
trinembinem
37
31
mag
38
32
woxbin
39
33
yontrin
40
34
kinbinam
41
35
lum
42
36
septrinbin
43
37
sil
44
38
levbinem
45
39
kintrinem
46
3A
fluorbin
47
3B
phos
48
40
trinbinom
49
41
sepem
50
42
kinembin
51
43
zottrin
52
44
yonbinem
53
45
sulf
54
46
trinambin
55
47
levkin
56
48
sepbinam
57
49
woxtrin
58
4A
nevbin
59
4B
chlor
60
50
kintrinbinem
61
51
gon
62
52
sodbin
63
53
septrinem
64
54
binix
65
55
yonkin
66
56
levtrinbin
67
57
tash
68
58
zotbinem
69
59
fluortrin
70
5A
sepkinbin
71
5B
cal
72
60
trinembinam
73
61
scan
74
62
magbin
75
63
kinemtrin
76
64
woxbinem
77
65
levsep
78
66
yontrinbin
79
67
teit
80
68
kinbinom
81
69
trinom
82
6A
lumbin
83
6B
van
84
70
septrinbinem
85
71
zotkin
86
72
silbin
87
73
nevtrin
88
74
levbinam
89
75
chrom
90
76
kintrinembin
91
77
yonsep
92
78
fluorbinem
93
79
sodtrin
94
7A
phosbin
95
7B
woxkin
96
80
trinbinum
97
81
mang
98
82
sepembin
99
83
levtrinem
100
84
kinembinem
101
85
fer
102
86
zottrinbin
103
87
cob
104
88
yonbinam
105
89
sepkintrin
106
8A
sulfbin
107
8B
nick
108
90
trinambinem
109
91
cup
110
92
levkinbin
111
93
magtrin
112
94
sepbinom
113
95
zing
114
96
woxtrinbin
115
97
fluorkin
116
98
nevbinem
117
99
yontrinem
118
9A
chlorbin
119
9B
zotsep
120
A0
kintrinbinam
121
A1
levem
122
A2
gonbin
123
A3
lumtrin
124
A4
sodbinem
125
A5
kinam
126
A6
septrinembin
127
A7
gal
128
A8
binixim
129
A9
siltrin
130
AA
yonkinbin
131
AB
ger
132
B0
levtrinbinem
133
B1
woxsep
134
B2
tashbin
135
B3
kintrinam
136
B4
zotbinam
137
B5
sen
138
B6
fluortrinbin
139
B7
sel
140
B8
sepkinbinem
141
B9
phostrin
142
BA
calbin
143
BB
yonlev
144
100
trinembinom
145
101
nevkin
146
102
scanbin
147
103
sepemtrin
148
104
magbinem
149
105
brom
150
106
kinemtrinbin
151
107
kryp
152
108
woxbinam
153
109
zottrinem
154
10A
levsepbin
155
10B
sodkin
156
110
yontrinbinem
157
111
rud
158
112
teitbin
159
113
sulftrin
160
114
kinbinum
161
115
fluorsep
162
116
trinombin
163
117
stron
164
118
lumbinem
165
119
levkintrin
166
11A
vanbin
167
11B
yit
168
120
septrinbinam
169
121
yonem
170
122
zotkinbin
171
123
woxtrinem
172
124
silbinem
173
125
zirk
174
126
nevtrinbin
175
127
sepkinem
176
128
levbinom
177
129
chlortrin
178
12A
chrombin
179
12B
niob
180
130
kintrinembinem
181
131
mol
182
132
yonsepbin
183
133
gontrin
184
134
fluorbinem
185
135
magkin
186
136
sodtrinbin
187
137
zotlev
188
138
phosbinem
189
139
septrinam
190
13A
woxkinbin
191
13B
teck
192
140
trinbinix
193
141
ruth
194
142
mangbin
195
143
yonkintrin
196
144
sepembinem
197
145
rhod
198
146
levtrinembin
199
147
pal
200
148
kinembinam
201
149
tashtrin
202
14A
ferbin
203
14B
nevsep
204
150
zottrinbinem
205
151
lumkin
206
152
cobbin
207
153
fluortrinem
208
154
yonbinom
209
155
woxlev
210
156
sepkintrinbin
As of 1202/03/01[z]=2018/03/01[d] I use:
ten,eleven = ↊↋, ᘔƐ, ӾƐ, XE or AB.
Base-neutral base annotations
Systematic Dozenal Nomenclature
Primel Metrology
Western encoding (not by choice)
Greasemonkey + Mathjax + PrimelDozenator
(Links to these and other useful topics are in my index post;
click on my user name and go to my "Website" link)
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SenaryThe12th
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Sep 6 2018, 11:20 PM #2

re #11, are you looking for things which have a natural ordering of some kind?  

I like this scheme; with only a few exceptions, it doesn't generate words which clash with pre-existing words, making them distinctive. 
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Kodegadulo
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Sep 7 2018, 12:17 AM #3

SenaryThe12th wrote: re #11, are you looking for things which have a natural ordering of some kind? 
Yes. 🙂
As of 1202/03/01[z]=2018/03/01[d] I use:
ten,eleven = ↊↋, ᘔƐ, ӾƐ, XE or AB.
Base-neutral base annotations
Systematic Dozenal Nomenclature
Primel Metrology
Western encoding (not by choice)
Greasemonkey + Mathjax + PrimelDozenator
(Links to these and other useful topics are in my index post;
click on my user name and go to my "Website" link)
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SenaryThe12th
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Sep 7 2018, 01:40 AM #4

Well I guess that rules out supermodels :-)   Names of the Popes or the British monarchs comes to mind, but those are obscure (although if Vice Presidents make the cut, maybe they might work as well).   Romanizations of chinese characters in unicode order would be practically inexhaustable, and be appreciated by a billion people....I wouldn't be one of them *chuckle*.   Books of the Bible, Surah's of the Koran,  Dewey Decimal Classes
https://en.wikipedia.org/wiki/List_of_D ... al_classes
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Kodegadulo
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Sep 7 2018, 02:47 AM #5

SenaryThe12th wrote:Well I guess that rules out supermodels :-)
Miss America pageant winners, on the other hand ... 🙂
Names of the Popes or the British monarchs comes to mind, but those are obscure (although if Vice Presidents make the cut, maybe they might work as well).
The problem with those is not so much obscurity as that such persons generally go by single given names, and an ordinal number when the few favorite names get recycled. There would not be enough variety to distinguish one from the other.  In the case of Popes, however, one might use their original pre-election names. In the case of royals, British or otherwise, this wouldn't help because they only came from certain royal houses and so share the same few surnames.
Last edited by Kodegadulo on Sep 7 2018, 03:55 PM, edited 1 time in total.
As of 1202/03/01[z]=2018/03/01[d] I use:
ten,eleven = ↊↋, ᘔƐ, ӾƐ, XE or AB.
Base-neutral base annotations
Systematic Dozenal Nomenclature
Primel Metrology
Western encoding (not by choice)
Greasemonkey + Mathjax + PrimelDozenator
(Links to these and other useful topics are in my index post;
click on my user name and go to my "Website" link)
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SenaryThe12th
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Sep 7 2018, 12:14 PM #6

Why not just programmatically generate the roots as well?    You could use a simple linear recurrence to generate random-looking syllables.
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Kodegadulo
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Sep 7 2018, 02:30 PM #7

SenaryThe12th wrote: Why not just programmatically generate the roots as well?    You could use a simple linear recurrence to generate random-looking syllables.
I might explore that idea, but I think that would actually be better utilized to create a completely different implementation of this concept.  In the implementation you're seeing here, it's just barely possible for someone to memorize the names of a lot of primes, based on vivid associations to chemical elements, presidents, vice presidents, etc. For instance, I've gotten to the point where I can associate 97d with manganese (mang), 101d with iron (fer), 103d with cobalt (cob), etc. But this is by no means the only possible implementation. I'd encourage anyone to try out whatever idea they like.  A more abstract implementation might be more amenable to automatic generation. However, it might be difficult for a human to memorize it, if it lacked any kind of allusion or association they could relate to.

One interesting possibility would be to pick phonemic elements that somehow mapped to the visual elements of UTL numerals, and use those to autogenerate the roots. For instance, UTL distinguishes primes whose modulo in base 6 are 1 vs 5, by using "rounded" shapes for the former, and "angular" shapes for the latter.  I could see that mapped, perhaps, to liquid/nasal consonants, versus plosive stops, respectively.
As of 1202/03/01[z]=2018/03/01[d] I use:
ten,eleven = ↊↋, ᘔƐ, ӾƐ, XE or AB.
Base-neutral base annotations
Systematic Dozenal Nomenclature
Primel Metrology
Western encoding (not by choice)
Greasemonkey + Mathjax + PrimelDozenator
(Links to these and other useful topics are in my index post;
click on my user name and go to my "Website" link)
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SenaryThe12th
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Sep 7 2018, 07:37 PM #8

If your interests are mnemonic, it might well be easier for somebody to memorize a simple algorithm than to memorize long tables of arbitrary names.
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Kodegadulo
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Sep 7 2018, 11:08 PM #9

SenaryThe12th wrote: If your interests are mnemonic, it might well be easier for somebody to memorize a simple algorithm than to memorize long tables of arbitrary names.
That is a plausible hypothesis, and you may very well be right. But people memorize the meanings of thousands of words in their own languages. While some of the grammar and morphology of various languages can be algorithmic, the arbitrariness of root words is anything but. Consider the monumental memorization feat the Chinese must accomplish to learn all the ideograms of their written language. I think you'd be hard pressed to squeeze those into an algorithm. I just have a hunch that it's actually easier to remember something as abstract as a prime integer if it can somehow be associated with a vivid mental image invoked by a name.

But it certainly would be interesting to put it to experiment. If we can devise the equivalent of UTL in phonemes, then we can compare how it scores in memorability against this more argam-like scheme.
As of 1202/03/01[z]=2018/03/01[d] I use:
ten,eleven = ↊↋, ᘔƐ, ӾƐ, XE or AB.
Base-neutral base annotations
Systematic Dozenal Nomenclature
Primel Metrology
Western encoding (not by choice)
Greasemonkey + Mathjax + PrimelDozenator
(Links to these and other useful topics are in my index post;
click on my user name and go to my "Website" link)
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SenaryThe12th
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Sep 11 2018, 01:55 PM #10

Kodo, it strikes me that if you were to adopt a signed senary representation, perhaps a-la proposal-for-signed-digit-representation-t1909.html, you would get a systematic naming scheme for every rational number which makes its prime factorization explicit.
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Kodegadulo
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Sep 11 2018, 04:47 PM #11

SenaryThe12th wrote: Kodo, it strikes me that if you were to adopt a signed senary representation, perhaps a-la proposal-for-signed-digit-representation-t1909.html, you would get a systematic naming scheme for every rational number which makes its prime factorization explicit.
You'll need to demonstrate that, because I just don't see it. How can any positional place value representation, in any base, signed or not, possibly make the prime factorizations of all natural numbers self-evident? Any base chosen will necessarily be a finite number, with a finite number of prime factors itself. Only those factors can contribute to what is self-evident in numbers represented in that base. But the potential number of primes that can show up in factorizations is infinite.
As of 1202/03/01[z]=2018/03/01[d] I use:
ten,eleven = ↊↋, ᘔƐ, ӾƐ, XE or AB.
Base-neutral base annotations
Systematic Dozenal Nomenclature
Primel Metrology
Western encoding (not by choice)
Greasemonkey + Mathjax + PrimelDozenator
(Links to these and other useful topics are in my index post;
click on my user name and go to my "Website" link)
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SenaryThe12th
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Sep 11 2018, 05:09 PM #12

 How can any positional place value representation, in any base, signed or not, possibly make the prime factorizations of all natural numbers self-evident? 
Well I wasn't talking about natural numbers, I was talking about rational numbers.  And I didn't say "self-evident", I said "explicit", by which I meant that it made the factorizations explicit in the same way that your proposed schemes makes the factorizations explicit for natural numbers.  
You'll need to demonstrate that,
Here's what I was thinking, currently bin <-> two, and trin <-> three.  Therefor, 6 would be trinbin.

For the powers, just use a signed senary number with the following suffexes encoding the digits:

-2 <->  -om
-1 <-> -um
0  <-> -ix
1 <-> im
2 <-> -em
3 <-> -am.

Then you could represent the rational number two-thirds as bintrinum.  You could approximate pi as twenty-two/seven, or levbinseptum, etc etc.  Or something like that.  I'm sure you could come up with a more euphonious scheme than I can.
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Kodegadulo
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Sep 11 2018, 06:58 PM #13

SenaryThe12th wrote:For the powers, just use a signed senary number with the following suffexes encoding the digits:

-2 <->  -om
-1 <-> -um
0  <-> -ix
1 <-> im
2 <-> -em
3 <-> -am.

Then you could represent the rational number two-thirds as bintrinum.  You could approximate pi as twenty-two/seven, or levbinseptum, etc etc.  Or something like that.  I'm sure you could come up with a more euphonious scheme than I can.
Oh I see. You weren't talking about the roots, you meant the exponents. Sure, we could find a way to represent negative exponents. But I don't think we need to do it by using a balanced signed base. We can just introduce a syllable that means "reciprocal". Let's say, -id:

bin over trin = trinidbin
trin over bin = trinbinid
un over trinbin = trinidbinid
levbin over sep = levsepidbin
binem over trinem = trinemidbinem
trinem over binem = trinembinemid
un over kinbin = kinidbinid
un over kinembinem = kinemidbinemid
un over trinbinem = trinidbinemid
un over trinembinom = trinemidbinomid
kinembinem over trinembinom = kinem over trinembinem = kinemtrinemidbinemid

But perhaps the "X over Y" expressions on the left are already good enough.
As of 1202/03/01[z]=2018/03/01[d] I use:
ten,eleven = ↊↋, ᘔƐ, ӾƐ, XE or AB.
Base-neutral base annotations
Systematic Dozenal Nomenclature
Primel Metrology
Western encoding (not by choice)
Greasemonkey + Mathjax + PrimelDozenator
(Links to these and other useful topics are in my index post;
click on my user name and go to my "Website" link)
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richard.chasen
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Sep 24 2018, 07:19 PM #14

      That idea is a spark of genius. I briefly looked it over and I'm impressed. 

I have one suggestion that will become important when humans have contact with other intelligent extraterrestial entities. Having a geometric based system for making numbers and then having names and roots based on this geometry. And yes my suggestion would be a difficult project. 
   I would hardly know where to begin a triangle is the best geometric represention for three, but what would one use for five (or any other number) cubed?
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