*The Most*,

*( Aside from The Gematria & Recursive Mandalas... )*

Is the Region from Zero to One!

You can put the whole Number Line in there,

Either just The Positive Numbers,

---

HP RPL Program Listing

**Converts Any Real Positve Number**

To it's 0 to 1 Equivalent

Input :

To it's 0 to 1 Equivalent

1: Any Number from Zero to + Infinity

1 + INV 1 SWAP -

*The Curious part here is the adding & subtracting of 1.*

*It would work fine for all numbers without that...*

*Except for Fractions Less than 1...!!!*

---

or Both The Positive & Negative Numbers...!

---

HP RPL Program Listing

**Converts Any Real Positve Number**

**To it's 0 to 1 Equivalent**

Input :

1: Any Number from - Infinity to + Infinity

DUP ARG COS SWAY OVER + INV - 1 + .5 *

*The Curious part here is ARG COS which replaces SIGN,*

Which fails with Zero...!

ARG COS returns 1 for 0, as it should!

---

Which fails with Zero...!

ARG COS returns 1 for 0, as it should!

This allows you to Graph Stuff without having to worry about

Numbers Falling Outside The Parameters of your Picture...!

---

One aspect of Zero to One is considering Just Zero & One.

On The HP48, The graphics

*( are normally )*either Black or White.

Included with The Calculator's Functions & Commands,

Are Two Graphic Operators...

*( Three; if you Include Replace ( ? ) )*

Graphic OR & Graphic XOR

*( Exclusive OR )*

With

**OR**; When you combine two graphics,

All The Black pixels on Graphic-A will remain Black

If The Pixel that overlays it, From Graphic-B, Is Also Black!

If The Pixel on Graphic A is White

& The Pixel on Graphic B is Black;

Then The Pixel will be Black!

If Pixel A is Black & Pixel B is White;

Then The Resulting Pixel will be Black!

If Both Pixels on Both Graphics

*( A & B*) are White;

Only Then will the Resulting Pixel be White.

This is normally shown Schematically like so:

-------------- Graphic A

-------------- Black -:- White

-Graphic B -----------:-------

---- Black ----Black -:- Black

---- White ----Black -:- White

Another way of writing this is :

**BBBW**

**Exclusive OR**works like this;

*Almost the Same*,

Except that Black & Black Makes White...!!!

-------------- Graphic A

-------------- Black -:- White

-Graphic B -----------:-------

---- Black ----White -:- Black

---- White ----Black -:- White

This would be written as :

**WBBW**

Since there are

*Two Options*with

*Four Variations*

There are

*16 Possible Ways*

To Combine The Two Graphical Pictures.

You may be wondering WHY

You would want to overlay two pictures in such a way

So that they either Reinforce or Cancel-Out

One or the others Colour...???

Amazingly; There are LOTS of Applications for this sort of thing...!

The most Notable is when you want to Fill In an Area

On a Graphical Picture with a Pattern.

To do this;

You First Make a copy of The Original Picture

and Set it Aside, [ Picture A ]

Then Fill in The Selected Enclosed Area with Black, [ Picture B ]

Then GXOR Picture A over Picture B and All the Pixels,

Except those in the Newly Filled Area will Magickally Disappear...!

Then Reverse that; So that White turns Black & Black turns White.

Then Create another Picture of the same size,

But contains nothing But the Pattern you Want, [ Picture C ]

Reverse This Picture as well.

Then Use GOR to combine these two pictures,

*And Reverse them.*

The Area that you chose will now have The Pattern in it...!

The last step is to GOR Picture A on top of this latest Picture,

And that will restore all the other Picture Elements

From the Original Picture...!

When you actually do this on a computer,

It's all done;

*'Behind the Curtains'*

So that when it's all done,

And suddenly POPS onto your screen,

It looks like it just happened without any effort at all...!!!

There are LOTS of other applications as well...

*But that's not what i'm interested in right now...*

i got to thinking some time ago

That there should be another Functional for

**Graphic AND**...

Only Black on Blacks would be Black,

*And all other Pixels would turn White!*

Then it further occurred to me

That there are still another 13 Possible Operations

Left Unaccounted for...!!!

At first i was thinking;

Many of these will require

*lots of fickling around...*

Such as

**The Right Handed GXOR**

That turns Black on White Pixels Black,

*While leaving White on Black Pixels White...!!*!

Black on Black turn White & White on White Stay White...!

BUT it turns out that

They are All Very easy & Simple to Construct...!!!

*Here's a list of them:*

-1 -WWWW- DROP OVER NEG GOR NEG - - - - - - - Null

The programs are all in RPL for the HP48 Series of Calculators,

The Input for each Program is the Same for Either GOR or GXOR or REPL

- -- 3: Graphic or PICT

- -- 2: (x,y) or { # xb #yb }

- -- 1: Graphic

-2 -WWWB- GOR NEG - - - - - - - - - - - - - - ~GOR

-3 -WWBW- SWAP ROT NEG GOR NEG - - - Graphic Minus

- - - - - - - - - - -3 is also The W on B AntiGXOR

-4 -WWBB- NEG 3 ROLLD DROP2 - - - - - - - - - - ~B

-5 -WBWW- NEG GOR NEG - - - - - - -B on W AntiGXOR

-6 -WBWB- DROP2 NEG - - - - - - - - - - - - - - ~A

-7 -WBBW- GXOR - - - - - - - - - - - - - - - -GXOR

-8 -WBBB- NEG SWAP ROT NEG GOR - - - - - - - ~GAND

-9 -BWWW- NEG SWAP ROT NEG GOR NEG - - - - - -GAND

10 -BWWB- NEG GXOR - - - - - - - - - - - - - ~GXOR

11 -BWBW- DROP2 - - - - - - - - - - - - - - - - -A

12 -BWBB- NEG GOR - - - - - - - - - - - -~AntiGXOR

13 -BBWW- 3 ROLLD DROP2 - - - - - - - - - - - - -B

14 -BBWB- SWAP ROT NEG GOR - - - - - - - ~AntiGXOR

15 -BBBW- GOR - - - - - - - - - - - - - - - - -GOR

16 -BBBB- DROP OVER NEG GOR - - - - - - -Tautology

As an Extension of this...

There are also 16 Fractional Propositional Logic Operators

That work on all The Intermediate Numbers

From Zero to One--

Rather than The Normal Zero OR One

For Ordinary Propositional Logic...!!!

These are also written for the HP48

So that Operators are First Laid Out

Then An Operation is Performed on Them...!

*Such that:*

[ p q + ] is equivalent to ( p + q ) anywhere else on the planet...!

*For ( 2 ) : 1 p q MAX -*

*Means : 1-(MAX(p,q))*

**Input for These Routines would be:**

- -- 2: - - - - p

- -- 1: - - - - q

Both p & q are Fractional Truth Values

Between & Including : Zero & One

**Output is A Fractional Value as Well...**

Signifying the Resultant Truth Value After The Operation.

*These should be consistent with*

*Conventional Truth Tables,*

*In so far as, Zeros & Ones are Concerned...!*

-1 -0000- 0 . . . . . . . . . . . . . . . .Null

Everything is Wrong : The World is Maya

-2 -0001- 1 p q MAX - . . . . . . . . . . . ~OR

Wrong is Right

-3 -0010- p 1 q - MIN . . . . . . . .I Said So!

p is Right, Unless q Agrees with p

-4 -0011- 1 q - . . . . . . . . . . . . . . .~q

q is Wrong

-5 -0100- 1 p - q MIN . . . . . . Just Because!

q is Right, Unless p Agrees with q

-6 -0101- 1 p - . . . . . . . . . . . . . . .~p

p is Wrrong

-7 -0110- p q - ABS . . . . . .Not Equal or XOR

Just so long as p & q Disagree, Then that is Right

-8 -0111- 1 p q MIN - . . . . . . . . . . . .~&

If p & q Both think that it's Right, Then it's Wrong

-9 -1000- p q MIN . . . . . . . . . . . . . . &

If Both p & q Think it's Right, Then it's Right!

10 -1001- 1 p q - ABS - . . . . . Equal or ~XOR

If Both p & q Agree, Either Right or Wrong, Then it's Right

11 -1010- p . . . . . . . . . . . . . . . . . p

pis Right

12 -1011- p 1 q - MAX . . . . . . . . . Because

p is Right, Because q is Right

13 -1100- q . . . . . . . . . . . . . . . . . q

q is Right

14 -1101- 1 p - q MAX . . . . . . . . . If Then

If p is Right, Then q is Right

*[ There is an Odd Bit here... It seems to imply That if some crazy thing*

*'p' is Given as False; Then Some Crazy thing q will be True...!*

*What it is Actually saying is that the Truth Values of p & q Determine*

*The Validity of the Operator... If p is False & q is True, The Operator is*

*Valid, But q is True for it's own reasons, Not because p is False.*

*This is perhaps the most compelling evidence that*

*Propositional Logic*

*Is Inherently Flawed*

*& Why the Aliens have not yet Overtly Contacted Us. ]*

15 -1110- p q MAX . . . . . . . . . . . . . .OR

When p or q is Right, Or Both are Right

16 -1111- 1 . . . . . . . . . . . . . Tautology

*Everything is Right with The World,*

*Everything is exactly what it is,*

*Everything that is exactly what it is, Is True,*

*Everything that is True, Is Beautiful*

*Bonus Feature...!*

PhotoEmulsion Propositional Arithmetic...!!!

PhotoEmulsion Propositional Arithmetic...!!!

With PhotoEmulsion Propositional Arithmetic,

You are only allowed to use the Physically Possible Tools

Of either Adding Two Photos Together by overlaying transparencies,

And Creating a Third Image ( Transitionally Negative )

Or Inverting an Image by Exposing it to The Proper Emulsified Plastic or Paper Material...

*Curiously;*When you Add Photographic materials together in this manner,

*The Results may be Somewhat CounterIntuitive...!*

When you add a 100% Black Transparency

To Another 100% Black Transparency,

The Result will be

*( After the Usually Negative Intermediate Stage )*

A100% Black Image

*...!!!*

When you add a 50% Black Image

To a 50% Black Image,

*The Result will be a 75% Black Image...!!!*

This is

*because*some of The Silver or Dye Pixels on the Emulsion

Will exactly overlay the pixels on the second image,

Nullifying their accumulative effect...!

It is simple to see that if Image A is 50% Covered with Pixels,

It will Block Out

*50% of the pixels*on Image B...

So that if Image A is 60% Black,

And Image B is 12% Black,

60% of Image B will be Nullified,

Leaving 40% of the 12%

To Add to the 60% of Image A...

Thus Creating an Image of 64.8% Blackitiness...!!!

*Likewise--*

9% + 16% = 23.56%...!!!

*Inverses;*On the other plate, Will be exactly the opposite,

*Just as you would expect them to be...*

...Pixel count wise...

60% Inverts to 40% -- And so on...

*This method of Inverting a Quantity is intended for Fractional Values Only...*

*So that 60% = .6 & 40% is the result of subtracting .6 from 1...! ( ? )*

*To Obtain the % Value; Multiply by 100...!*

[

*As a harmless Digression.*..

This is The Third Way to

*Oppositify*a Number...!

Method One ) The Opposite of 3 is -3

Method Two ) The Opposite of 3 is .333...

Method Three ) The Opposite of .3 is .7

Method Four ) The Opposite of 30 is 210

*Method Fourb ) The Other Opposite of 30 is 330 !*

Method Five ) The Opposite of 3 is...???

]

+

*means*

**PhotoEmulsion Addition**

. . . HP RPL Version : p q + p q MIN p q MAX % 100 * -

~

*means*

**PhotoEmulsion Inversion**

. . . HP RPL Version : 1 p -

-1 -0000- 0 . . . . . . . . . . . . . . . Null

-2 -0001- p q + ~ . . . . . . . . . . . . .~OR

-3 -0010- p ~ q + ~. . . . . . . . .I said So!

-4 -0011- q ~ . . . . . . . . . . . . . . . ~q

-5 -0100- p q ~ + ~. . . . . . . Just Because!

-6 -0101- p ~. . . . . . . . . . . . . . . .~p

-7 -0110- p ~ q + ~ p q ~ + ~ +. . . Not Equal

. . . . . . . . . . . . . . . . . . . . or XOR

-8 -0111- p ~ q ~ + . . . . . . . . . . . . ~&

-9 -1000- p ~ q ~ + ~ . . . . . . . . . . . .&

10 -1001- p ~ q + ~p q ~ + ~ + ~ . . . . Equal

11 -1010- p . . . . . . . . . . . . . . . . .p

12 -1011- p q ~ +. . . . . . . . . . . Because

13 -1100- q . . . . . . . . . . . . . . . . .q

14 -1101- p ~ q + . . . . . . . . . . .If Then

15 -1110- p q + . . . . . . . . . . . . . . OR

16 -1111- 1 . . . . . . . . . . . . .Tautology

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*Extra Bonus:*

Is there a way to Codify

*Fractional Propositional Logic*

For a Mechanical System;

Such as Techno-Lego Blocks ( And Accessories )

or Tinker Toys...???

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