Over the ☽

$$\rightmoon ^\circ =\rightmoon$$

Welp.. I think I’ve officially gone loony! 😅

I can’t stop thinking about the moon.. I learned about $\circledcirc$ (sunny) & $\rightmoon$ (loony) at the same time:

$$\circledcirc=\{0*, 1*, 2*,3*,...\}$$

$$\rightmoon+0=\rightmoon+*1=\rightmoon+*2=...=\rightmoon+\rightmoon=\rightmoon$$

The interesting thing to me is that since $\rightmoon +n=\rightmoon$ we have $\rightmoon +\circledcirc =\rightmoon$. It may not be that interesting, but I think it’s a beautiful equation.

Additionally, we have $\rightmoon n=\rightmoon$, which leads to another beautiful equation:

$$\rightmoon ^ {\rightmoon} = \rightmoon$$

A part of what has me in lunatic mode are 2 papers I recently found:

In the first paper the authors introduce affine impartial / normal play games.

While most of the paper is above my head, it did inspire an idea!

Can $\infty$ in affine impartial ( $\mathbb{I}m^\infty$) & normal play ( $\mathbb{N}p^\infty$) be replaced w/ $on$ to form $\mathbb{I}m^{on}$ & $\mathbb{N}p^{on}$?

Affine games do not include transfinite nimbers (but they can produce infinitely many finite nimbers). I think replacing $\infty$ w/ $on$ would result in a class of games that does include transfinite nimbers. I’m just not sure what, if anything, would break..

My basic line of thinking of why this might work is:

  • $on=\{on|\}\in \mathcal{L}$ (right has no moves & left can pass) (1.1)
  • $off=\{|off\}\in \mathcal{R}$ (left has no moves & right can pass) (1.2)
  • $\forall x \in \mathbb{I}m^{on}\backslash\{off\},on+x=on$ (1.3)
  • $\forall x \in \mathbb{I}m^{on}\backslash\{on\},off+x=off$ (1.4)
  • $on+off=dud$ (1.5)
  • $0=\{off|on\}$

I’m not sure how to evaluate $\{on|on\}$ or $\{off|off\}$. I might however naively call these $on*$ & $off*$.

I playfully would call these projective games.

Oh right.. the big kicker in all of this? $\rightmoon$ is equal to the set of no nimbers. That is:

$$\rightmoon = \{\}$$

#NES to the max 💯

What I keep wondering is.. is $\rightmoon$ “the” master idempotent?? 🤔 That which has the maximal absorbing nature (compared to let’s say the adsorbent nature of the traditional empty set $\emptyset$).


In other news, after placing a bounty on a question about a multiplication nimonic (nimber mnemonic), I got an amazing answer! I initially randomly stumbled upon a simple multiplication nimonic after seeing 1 for addition. This led me to create a larger nimonic, which then led to trying to find the optimum configuration.

Enter PG(3,2). There is a nice 3d graph that I have been considering updating with labels for the given solution. I started playing around with three.js as well (peep the new hypersurreal homepage).

Need. Coffee. TTFN

~StarGirl 👩‍🎤

  • comment

Comments

I forgot to add the Moonlight Theorem 🤦‍♀️

When a moon appears as a part of a component, the value of the component as a whole is also equal to the moon (a loony move exposes a loony game)

It’s like some sort of black hole!

Add a comment

You must sign in to post a comment.