Find a surface that is an immersion of $\Bbb RP^2$ in $\Bbb R^3$ based on the algebraic curves

This question is motivated by the Lorenz curve used in economic analysis and also the Penrose diagram used in general relativity, used by physicists in order to visualise causal relationships in compactified Minkowski space time models.

It is also motivated deeply by Hamiltonian mechanics, symplectic geometry, and contact geometry (which can be viewed as the odd dimensional counterpart of symplectic geometry). Hamiltonian mechanics was a reformulation of Newtonian mechanics. In Hamiltonian mechanics one studies phase spaces of physical systems, symplectic flows and many other topics.

From a mathematical standpoint, this question is motivated from modern trends in differential geometry, topology, abstract algebra and mathematical physics.

Consider the families of planar curves for $$\Re(s) \ge1:$$

$$\zeta:= \{ (x, y) \in \Bbb R^2 | x^s + y^s = 1 \}$$ $$\tau:= \{ (x, y) \in \Bbb R^2 | (1-x)^s + y^s = 1 \}$$ $$\psi:= \{ (x, y) \in \Bbb R^2 | x^s + (1-y)^s = 1 \}$$ $$\phi:= \{ (x, y) \in \Bbb R^2 | (1-x)^s + (1-y)^s = 1 \}.$$

Interpreting $$\zeta,\tau,\psi,$$ and $$\phi$$ as phase spaces allows the interpretation of them as infinite dimensional manifolds, specifically infinite dimensional symplectic manifolds:$$(\zeta,\omega),(\tau,\omega),(\psi,\omega),(\phi,\omega).$$

What follows is a natural Hamiltonian vector field, which defines a Hamiltonian flow on each of the symplectic manifolds.

The process of lifting these manifolds into $$\Bbb R^3$$ can be achieved with homotopic maps. See the answer by Paul Frost here: https://math.stackexchange.com/questions/2895816/existence-of-homotopic-map. The key thing to understand is that each of the four symplectic manifolds are unique projections of the curves lifted via homotopy. In other words, one can project the lifted curves onto the planar curves bijectively. The image provides the intuition for what a single lift looks like. But in reality there are infinitely many of these lifts each at different heights above the planar curves.

Questions $$($$choose one$$)$$:

$$1)$$ Can one glue together $$\zeta,\tau,\psi,\phi$$ including all lifts and treat the whole thing as a manifold? What would be the full classification of this manifold?

$$2)$$ Can a $$3$$-manifold be constructed, whose geodesics project to these algebraic planar curves?

I think yes, because one can project all lifted phase spaces down to the corresponding planar phase spaces. I’m pretty sure the $$3$$-manifold is a submanifold of the infinite dimensional manifold.

$$3)$$ Define a surface that is an immersion of $$\Bbb RP^2$$ in $$\Bbb R^3,$$ that looks like this, and is a three dimensional analog of the two dimensional equations above:

This is a shape I built to try to get an intuitive understanding of the geometry of the manifold and phase space in three dimensions: The Boy’s Surface is an example of an immersion of $$\Bbb RP^2$$ in $$\Bbb R^3.$$

The image relates to the equations because each white strand in the image corresponds to a strand of a geodesic on a $$2$$-manifold, namely $$S^2.$$ Each white geodesic also corresponds to a specific planar curve listed above for a specific value of $$s$$. In the picture $$s=2.$$ The singularities can be viewed as invariants or fixed points because they don’t change spatial location for the purpose of this question. Varying $$s$$ gauges the way the shape looks. As $$s$$ approaches infinity the shape will look like a cube. As $$s$$ tends to $$1,$$ the shape will look like $$2$$ pairs of perpendicular lines situated in $$3$$-space.

Maybe it helps to say that the image is the shape of the intersections of $$8$$ identical copies of $$S^2$$ with finitely many geodesics shown. It is like a higher dimensional venn diagram if you will. Suggestions for decreasing metagaming and increasing player immersion?

This question is posed in a great way over on the Paizo boards, where I hang out since I play/run a lot of Pathfinder. Immersion is one of the key parts of the RPG experience to me, so I loved the question, and wanted to open it up to this community and also generalize it to other RPGs.

(Quick clarification – “immersion” in the sense of “players take on the roles of their characters in the game world as much as possible”. We used to call that “roleplaying” till the term got co-opted, and now some folks are trying to use immersion in different senses, like “engrossed in the story”. This question is only about in-character immersion.)

What are your favorite techniques – as a GM or even as a player – for promoting and maintaining character immersion (aka “roleplaying your character”) over metagaming? Where metagaming is “I know what I rolled,” “I know all the monsters in the manual,” “In 90% of plots this guy would be the bad guy,” or other things that should properly be outside the game fiction? I’ve tried to do this as a GM but also struggled with it as a player; I resent it when I feel like I’m “forced to metagame” by the scenario or GM to keep the adventure going.

In the OP, there’s a lot of focus on making rolls behind a screen, especially skill checks and saving throws and the like, or having the GM track hit points instead of the players. Some of that works well, but in practice could overwhelm the GM and disenfranchise players who want some sense of “ownership” over their characters. I’d like to hear techniques people have actually used (not untried opinions) and how well they worked and what their side effects were.

To set a good example, here’s one answer from me – I ran a multiyear campaign where as GM I practiced strict information compartmentalization – I didn’t say things in front of the group that only a subset of the party witnessed. I passed notes and took people aside. This worked very well in terms of helping people immerse and keep a realistic in-world viewpoint. But it did slow the game down, especially once things got more interactive and the player was writing multiple notes back for clarification, or a “take half the party aside” turns into 30 minutes of action excluding the other characters. I tried to mitigate that by going back and forth to spread spotlight time, but sometimes one group would just say “We sit here and wait for them to be done with whatever the hell they’re doing…”

Posted on Categories Best Proxies

Is Cohen immersion conjecture (theorem) known for vector spaces?

R. Cohen proved the immersion conjecture in a 1985 Annals paper:

Any smooth compact n-dimensional manifold admits an immersion into Euclidean space of dimension 2n-a(n), where a(n) is the number of 1’s in the binary decomposition of n.

Is there any result of this kind for vector bundles E over compact manifolds? Notice that the sphere bundle of E is compact. Maybe there is a silly argument…

Closed immersion inducing isomorphisms on stalks

Does there exist a closed immersion of schemes that induces isomorphisms on stalks, has a non-empty source, an irreducible (possibly non-reduced) target, and is not an isomorphism? Such $$f$$ is automatically flat, so it would have to be non-surjective. If $$f$$ were locally of finite presentation, it would be open and thus surjective so it can not be locally of finite presentation.

Assume that definitions are those given in Stacks project.

Glueing a locally closed immersion from locally closed immersions on fibers

Let $$X$$, $$Y$$ be two irreducible Noetherian schemes. Assume that for each prime ideal of $$\mathbb{Z}$$ $$I$$ we have a locally closed immersion $$f_I:X\times \mathrm{Spec}\,k(I)\rightarrow Y\times \mathrm{Spec}\, k(I)$$. When does there exist a locally closed immersion $$f:X\rightarrow Y$$ base changing to $$f_I$$?

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What less-loaded, less-ambiguous terms exist that are equivalent to ‘associative and dissociative mechanics’, ‘immersion’ etc.?

This question has arisen due repeatedly facing misunderstandings when trying to discuss certain differences in the handling of mechanics, decisionmaking etc. between RPGs or between different parts of a single RPG. So first, here is the description of the two contrasting cases in abstract terms, hopefully as long-winded as necessary to ensure clarity (if necessary, it will be edited and expanded).

Definition in Abstract Terms

• Pattern A: The situation as perceived by the PC is described by the GM. The player decides how the PC decides to react to the situation, and describes the PC’s action. In accordance to this action of the PC, a game mechanic is used to resolve further outcomes. The mechanic in question is an outcome of the PC’s decision and subsequent action. In a perfect example, the choice of which mechanics to employ hinges solely and unambiguously on the description of the PC’s thoughts and actions, and can be derived from them even if the player didn’t convey system information (but in practice most examples aren’t this perfect).

• Pattern B: The situation as perceived by the PC is described by the GM. The player decides what outcomes the player wants and what mechanics can help achieve that. In accordance to this player decision, a game mechanic is used to resolve further outcomes. The mechanic in question is fully independent of the PC’s decision and subsequent action, and the decision to use the mechanic does not represent any decision of the PC. In a strong example, whether the mechanic is to be used cannot be derived from the sole description of the PC’s thoughts and actions, and requires the player to speak in system terms (in practice not all examples are like this, but many are).

Less Abstract Examples

• Pattern A: The GM describes a skeleton monster as dangerous but flimsy. The mighty warrior decides to forego defence and hit the skeleton with all his strength, hoping to slay it in one blow. The game mechanics reflect the warrior’s decision in the form of giving the All-Out Attack bonus to damage (which negates Active Defences). The player rolls the dice to see if the warrior hits.

• Pattern B: The GM describes said mighty warrior’s swing as going off-target (alternatively, the player may just see the negative outcome on the dice and react without waiting for the GM’s description). The player decide that in this case, failure is unacceptable, and decides to use the Luck trait, which allows making a reroll. The warrior did not decide to be lucky this moment instead some other moment – the decision was strictly on the player’s head. The reroll mechanic is used strictly based on the player’s decision.

Terms I’ve Seen Used to Describe Pattern B, and Why I Seek Alternatives to Them

• Supernatural: One that I think least used and also most misleading, and yet I have encountered a few times. From an inside-the-campaign-world perspective, nothing supernatural actually happened in Pattern B. In fact a lot of the time Pattern B assumes that all mechanically possible outcomes are supposed to also be naturally possible in-setting.

• Meta: People object to it as meaningless by statements such as ‘Rolling the dice is a metagame action! It’s not something characters do after all’.

• Narrative, Narrative Control etc.: generally using the word ‘narrative’ in discussion of RPGs tends to result in people jumping to the conclusion that I must be using it in a GNS meaning and cannot ever possibly be meaning anything else.

• Immersion-breaking (contrasting to Pattern A being labelled immersive): people usually object by saying ‘Thinking about whether to accept a Compel or declare a Story Detail doesn’t break my immersion, but counting HP does!’, which at best results in a long and repetitive discussion of what exactly is meant by the word ‘immersion’ by a given person and how the definitions radically differ.

• Dissociative Mechanic (contrasting to Pattern A being labelled associative): people also object to it in a similar manner – ‘I don’t feel dissociated from my character when I think of the ways my warrior could get lucky, I do when I calculate encumbrance’; another objection I seen was a claim that the term was coined with the goal of being derogatory of such mechanics (a belief I do not share but am losing any enthusiasm to dispute by now).

• Author Stance (contrasting to Pattern A being labelled Character/Actor Stance): the objection is generally along the lines of ‘Oh, you called me an Author, as if I’m not an actual roleplayer / not engaging in roleplaying, but rather just writing a book’. I there are some roots of this objection in the reference to the advice for railroading GMs to write a book instead, though it seems like a stretch.

So given the flaws of the above terms, what would be a good replacement? I’m seeking terms that are unambiguous, intuitive and not prone to causing definition arguments and flame wars. Yes, I know I won’t get a perfect, 100% safe solution, but I’m seeking something that is at least better than the ones I found so far.

Posted on Categories Best Proxies