Challange: How Ugly Can you make your favorite base?

SenaryThe12th
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SenaryThe12th
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Joined: Mar 1 2018, 02:03 PM

Mar 7 2018, 02:26 AM #1

New challenge:  how ugly can you make your favorite base?  The idea is to
take a beautiful base, and destroy as many of its good properties as possible
by combining the various techniques found on this formum.   Here's my entry
in the contest.

As you can tell by my moniker, I'm a big fan of base 6.  I'm also a big fan
of twistaff notation, and ballanced bases.  But what happens when you put them
all together?  Do you get easier divisibility tests, nice rounding, and
straightforward conversion between positive and negative numbers?   Uh, no.


We'll do a 3-on-2 version of twistaff senary.  But for an extra twist,
we'll use ballanced ternary for the upper stave:
decimal balanced ternary
-1 I
0 0
1 1
     

and we'll use straight binary for the lower stave.  Standard tables
don't look too bad....

Code: Select all

Addition:
----------

       Upper staff        Lower staff
        -----------        -----------

        |  I  0   1          |  0  1
     ---+----------        --+------
      I | I0  I   0        0 |  0  1
        |                    |
      0 |  I  0   1        1 |  1 10
        |
      1 |  0  1  10


Multiplication
--------------


       Upper staff         Lower staff
       -----------         -----------

        |  I  0  1           | 0   1
     ---+---------         --+-----
      I |  1  0  I         0 | 0   0
        |                    |
      0 |  0  0  0         1 | 0   1
        |
      1 |  I  0  1





... but here's how the numbers -12 through 12 decimal come out in this scheme:

Code: Select all

  dec     formula            twistaf notation
     --------------------------------------------

     -12     (2*1 + 1) * 6^0     I0.00
             (2*I + 0)

     -11     (2*1 + 1) * 6^0     I0.01
             (2*I + 0)

     -10     (2*1 + 1) * 6^0     I0.10
             (2*I + 0)

      -9     (2*1 + 1) * 6^0     I0.11
             (2*I + 0)

      -8     (2*1 + 0) * 6^0     I1.I0
             (2*I + 0)

      -7     (2*1 + 1) * 6^0     I1.I1
             (2*I + 0)

      -6     (2*0 + 0) * 6^0     I1.00
             (2*I + 1)

      -5     (2*0 + 1) * 6^0     I1.01
             (2*I + 1)

      -4     (2*1 + 0) * 6^0     I1.10
             (2*I + 1)

      -3     (2*1 + 1) * 6^0     I1.11
             (2*I + 1) * 6^1

      -2     (2*I + 1) * 6^0     00.I0


      -1     (2*I + 1) * 6^0     00.I1


       0     (2*0 + 0) * 6^0     00.00


       1     (2*0 + 1) * 6^0     00.01


       2     (2*1 + 0) * 6^0     00.10


       3     (2*1 + 1) * 6^0     00.11


       4     (2*I + 0) * 6^0     01.I0
             (0*0 + 1) * 6^1

       5     (2*I + 1) * 6^0     01.I1
             (2*0 + 1) * 6^1

       6     (2*0 + 0) * 6^0     01.00
             (2*0 + 1) * 6^1

       7     (2*0 + 1) * 6^0     01.01
             (2*0 + 1) * 6^1

       8     (2*1 + 0) * 6^0     01.10
             (2*0 + 1) * 6^1

       9     (2*1 + 1) * 6^0     01.11
             (2*0 + 1) * 6^1

      10     (2*I + 0) * 6^0     10.I0
             (2*1 + 0) * 6^1

      11     (2*I + 1) * 6^0     10.I1
             (2*1 + 0) * 6^1

      12     (2*0 + 0) * 6^0     10.00
             (2*1 + 0) * 6^1




MY EYES!!! MY EYES!!! its so ugly it burns.   How ugly is it?  Let us
count the ways:


1.  Yeah, counting is super hard.  My favorite is going from -2 to -3.
    Which implies we've taken senary's legendary prolixity and put it
   

Which implies we've taken senary's legendary prolixity and put it
    on steroids.  -3 takes 4 staves to represent, and parsing it to
    find the actual value is a challange in itself: What is the value
    of I1.11?  lets see, that would be -12 + 6 + 2 + 1 = -3.



2.  Which implies we've taken senary's legendary prolixity and put it
    on steroids.  -3 takes 4 staves to represent, and parsing it to
    find the actual value is a challange in itself: What is the value
    of I1.11?  lets see, that would be -12 + 6 + 2 + 1 = -3.


3.  We've managed to destroy the ballanced base property that there are
    multiple representations for each number.   Sounds like a good thing;
    but we have also thereby destroyed cleverly exploiting that ambiguity
    to be able to give ourselves banker's rounding by truncation.  Rounding
    in this notation mindbendingly difficult, at least for me.


4.  We've also managed to destroy the balanced base property that
    the negation of a number is a simple swap of positive for negative
    staves.  1 is 01, but -1 is I1.    Hideous!


5.  Is criss-cross multiplication at least any easier?  How
    hard can it be to just be multiplying by zero or plus/minus
    one all the time?  Well, the "pair-triple" table, or whatever
    you call it, which is analogous to the dicker-dozen table
    in twelfty, actually has an entry with a carry to overflow.
    So no, criss cross multiplication is totally screwed as well.

Code: Select all

 "Dicker Dozen"
         ------------

        |   I    0    1
      --+--------------
      0 |  0I   00   01
        |
      1 |  10   11  100 <----ugly!!!

       
6.  We have, however, managed to preserve all of the difficulties
    inherant in doing long division in ballanced bases.


Well, my work here is done.  I can't think of any more ways to
to make senary harder to work with.  Can you?
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Einmaleins
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Einmaleins
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Joined: Feb 12 2018, 06:58 AM

Mar 7 2018, 08:33 AM #2

Fascinating.
But I do not understand one word of this!
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Kodegadulo
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Kodegadulo
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Joined: Sep 10 2011, 11:27 PM

Mar 7 2018, 11:28 AM #3

Okay, we get it, Sen. You think senary is beautiful. Okay, use it and be happy. You think balanced bases and mixed/alternating bases are ugly (despite methinks thou dost protest too much the contrary). So don"t use those techniques, and be happy. You think it absurd to combine techniques you find ugly? Who is proposing such combinations, besides yourself? Are you just here to complain and provoke flame wars? Hmm, I think the Internets have a word for that...

It's customary for newcomers to the forum to post a little introduction about themselves in the Getting to Know You section. Why don't you introduce yourself and your interests, so we can properly welcome you?

P.S. I can't think of anything that makes a post look uglier than manually putting hard line breaks inside a paragraph at the point where you think the margin is, on your device, when both the post editor, and the final html rendering, do automatic line wrapping responsively, based on the widths of everyone's devices.
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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SenaryThe12th
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Joined: Mar 1 2018, 02:03 PM

Mar 8 2018, 02:42 AM #4



*chuckle* you took the post far too seriously.  But I have to admit, it does my heart good to hear you defending twistaff notation and I heartily agree with the "use your favorite number base and be happy" vibe.  

The post was basically my way of consoling myself for a failed project.   I do like senary, and I do love twistaff notation, and I do love balanced bases.  So I tried to combine all three of them to get a SUPER DUPER base and notation which would combine the best aspects off all three.

Imagine my disappointment when I found out that it was kind of thinking that "Hey, I love pickles, and I love ketchup, and I love ice cream.  How cool would it be to combine all three of them together?????"

So I posted this in the hopes that somebody would at least get a few laughs at my expense. 
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SenaryThe12th
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SenaryThe12th
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Joined: Mar 1 2018, 02:03 PM

Mar 8 2018, 03:19 AM #5

Einmaleins wrote: Fascinating.
But I do not understand one word of this!
Well, I have good news and bad news :-)   Good news is that somewhere, on this very forum, there are wonderfully written descriptions of everything I mention here.  Seriously good stuff, sometimes written by the very inventors of the techniques themselves, and almost always full of enlightening and mind-blowing insights.

Bad news is that the forum just migrated to a new hosting platform, so a lot of the connecting links between the articles have been lost.  And some of the formatting has gone to hash as well.   So putting it all together is going to be hard for a while :-(  But don't let that discourage you.  And if you ask enough questions, it might motivate some of those guys to rewrite/rebuild the stuffs.
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Kodegadulo
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Kodegadulo
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Joined: Sep 10 2011, 11:27 PM

Mar 8 2018, 03:39 AM #6

Well it's good and it's bad, but it's hardly news. We've been warned about this impending tapatalk migration for months. As for the garbaging of posts, some of that can be ameliorated: see my recent  Userscript(s) to Fix Tapatalk thread.
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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Shaun
Dozens Disciple
Shaun
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Joined: Aug 2 2005, 04:09 PM

Mar 8 2018, 07:02 AM #7

Luckily we have members here - especially Kode - capable of fixing some of the problems caused by the migration; and I am in touch with the Tapatalk co-ordinator who has their team looking into (and with any luck fixing) things.
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Einmaleins
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Einmaleins
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Joined: Feb 12 2018, 06:58 AM

Mar 8 2018, 12:20 PM #8

SenaryThe12th wrote:

*chuckle* you took the post far too seriously.
So I posted this in the hopes that somebody would at least get a few laughs at my expense. 
You like being laughed at?  Well, each to their own.
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Einmaleins
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Einmaleins
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Joined: Feb 12 2018, 06:58 AM

Mar 8 2018, 12:26 PM #9

SenaryThe12th wrote:

Good news is that somewhere, on this very forum, there are wonderfully written descriptions of everything I mention here.  Seriously good stuff, sometimes written by the very inventors of the techniques themselves, and almost always full of enlightening and mind-blowing insights.
Mind-blowing and enlightening? Well, these are also very peculiar ideas and histories that some odd have posted and caused hot discussions.
And a few diatribes. It is a strange world. Fascinating.
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icarus
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icarus
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Joined: Apr 11 2006, 12:29 PM

Mar 8 2018, 02:30 PM #10

I wrote a lot of the senary description. Double sharp also wrote quite a bit about it. I don't have time today to talk because of deadlines, but will respond in time. I don't think it's necessarily trolling, just hubris and confidence. Nothing wrong with the latter. My question would be (and I can't reply to it right away) do you use senary often in daily life? How? I'll talk to ya.
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SenaryThe12th
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SenaryThe12th
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Joined: Mar 1 2018, 02:03 PM

Mar 8 2018, 08:42 PM #11

icarus wrote: I wrote a lot of the senary description. Double sharp also wrote quite a bit about it. I don't have time today to talk because of deadlines, but will respond in time. I don't think it's necessarily trolling, just hubris and confidence. Nothing wrong with the latter. My question would be (and I can't reply to it right away) do you use senary often in daily life? How? I'll talk to ya.
Thanks for giving me the benefit of the doubt.  It wasn't trolling, nor was it confidence.  It was supposed to be kind of a joke on myself.  I tried to create a SUPER DUPER base + notation by throwing in the kitchen sink.   Take senary, mix in balanced bases, with a dash of twistaff notation, and I thought I could create a system which:
1.  had the great divisibility properties of senary
2.  had the nice rounding by truncation properties of balanced bases, and
3.  had the dirt simple multiplication--only ever had to multiply by -1, 0, or 1.

I guess it was hubris to think I could create something like that :-)

Because when I put it all together, the result was this frankenstein monster that not even a mother could love.  Instead of getting the best of 3 worlds, I had systematically destroyed everything which made any of the techniques desirable when used in isolation :-)

I was bummed for a while, but then I started laughing because I thought the joke was on me.  And I thought others might appreciate a cautionary tale.   What's more, I actually do think it might be a fun challenge:  take your favorite base, and see just how ugly you can make it.  I certainly learned a lot about balanced bases, twistaff notation, etc while doing the exercise, and now am sadder by wiser about the use and abuse of these techniques.

I did not expect to be accused of trolling and dumping poop on everybody's favorite stuffs :-( :-(   Geez, you guys are like suffering from post-traumatic stress or something.  Ya'll need to lighten up.  Not everybody in the world is a bitter old troll.
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SenaryThe12th
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Mar 8 2018, 11:08 PM #12

Einmaleins wrote:
You like being laughed at?  Well, each to their own.
Dude, if I wasn't able to laugh at a joke even when I was the butt of it, I would miss out on some of the greatest jokes in the world.
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Shaun
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Shaun
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Joined: Aug 2 2005, 04:09 PM

Mar 9 2018, 07:19 AM #13

"As you can tell by my moniker, I'm a big fan of base 6. "
Senary the 12th - so "Enery the 8th" perhaps?
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SenaryThe12th
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Mar 9 2018, 10:59 AM #14

Shaun wrote: "As you can tell by my moniker, I'm a big fan of base 6. "
Senary the 12th - so "Enery the 8th" perhaps?
I yam I yam :-)
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