trefoilknot on 5/9/2020 at 19:16

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Hi all,

FenPhoenix recently posted a (https://www.youtube.com/watch?v=W0m88sziq1o) Let's Play of my latest mission, (https://www.ttlg.com/forums/showthread.php?t=150533) Triptych on Infinity, Act I: Heist at Hilbert's Highrise Hotel.

Watching his play-through inspired me to make a post explaining the history and mathematics which inspired the mission. This post contains spoilers, so I don't recommend reading on if you have not yet played the mission, and want to be surprised!

Ive included a number of wikipedia and youtube links, for those who are curious. Hope you enjoy!

Characters:

The primary antagonists are Hilbert and Cantor (a.k.a., "Cantor the Corruptor"). Cantor is a deceased and disgraced former archmage of the Hand Brotherhood (his resignation letter quoted in the intro cutscene). His ideas were considered too dangerous by his rivals within the Brotherhood, Kronecker and Poincare, who now seek to prevent his ideas from posthumously infecting the world. David Hilbert owns the hotel and seeks to discover the truths Cantor left behind. All characters are loosely based on the real-life mathematicians after which they are named.

(https://en.wikipedia.org/wiki/Georg_Cantor) Georg Cantor was a 19th century mathematician who invented set theory. He is arguably most famous for demonstrating the uncountability of the "real" numbers (numbers from anywhere on the number line). That is, he proved that there are more real numbers than there are "natural" numbers (1, 2, 3,...). The same is not true of rational numbers (i.e., numbers which can be expressed as a ratio of two integers)---there are exactly as many rational numbers as there are natural numbers.

Many of Cantor's mathematical contemporaries considered his ideas to be, at best, ridiculous and, at worst, heresy. Many mathematicians of the time considered infinity and God to be inextricably linked ideas. The idea that there might be multiple differently-sized infinities was tantamount to pantheism, and therefore in conflict with the first of the Ten Commandments. Cantor vehemently rejected this notion, but was nonetheless ridiculed mercilessly with the bulk of the abuse coming from mathematicians Leopold Kronecker and Henri Poincare. Cantor suffered severe bouts of madness/depression, ultimately dying in an asylum. While much of his mental illness can be attributed to the abuse he received from his peers, he also addled his mind contemplating the question: "Is there a set whose cardinality lies between that of the natural numbers and the reals?" In other words, is there a set larger than the natural numbers, but smaller than the real numbers? He did not believe there was one- a notion which came to be known as the (https://en.wikipedia.org/wiki/Continuum_hypothesis) "Continuum Hypothesis." Over a century later, this problem remains unsolved. However, it has been proven that answering this question is not possible within our existing mathematical infrastructure (ZFC set theory).

(https://en.wikipedia.org/wiki/David_Hilbert) David Hilbert was one of the first influential mathematicians to see value in Cantor's work on set theory. He popularized Cantor's ideas, in part by using his power as editor of the top Mathematics journal. Within a generation, Hilbert brought Cantor's views into the mainstream. Hilbert's contributions were numerous. One of his many inventions (or discoveries, depending on how you conceptualize mathematics) is the space-filling (https://en.wikipedia.org/wiki/Hilbert_curve) "Hilbert Curve," a one-dimensional curve which can entirely fill a 2-dimensional space. By extension, it shows that information in any higher dimension can be losslessly encoded in a lower dimensional space.
(https://www.youtube.com/watch?v=3s7h2MHQtxc) Here's a fun/informative video about Hilbert Curves.



Setting: Hilbert's Highrise Hotel

Hilbert's Highrise Hotel is an infinite hotel with [countably] infinitely many floors and guest rooms. (Try running all the way up the stairs or jumping off the roof!)

https://i.imgur.com/UERWltn.png

The inspiration comes from "( https://en.wikipedia.org/wiki/Hilbert%27s_paradox_of_the_Grand_Hotel) Hilbert's Paradox of the Grand Hotel," a thought experiment devised by mathematician David Hilbert regarding a hotel with [countably] infinitely many guest rooms.

The thought experiment is as follows: consider a hotel with [countably] infinitely many guest rooms, all occupied. If a new guest shows up, can they be accommodated? Yes, simply move the guest in room k to room k+1 for all natural numbers k=1,2,3,... this will free up room 1 for the new guest while still accommodating the existing guests.

Further suppose [countably] infinitely many new guests show up- can they all be accommodated? Again, yes. Simply move each guest from room k to room 2*k for all natural numbers k=1,2,3,... Now there are infinitely many odd-numbered rooms available for the new-comers while still accommodating the existing guests. Etc...

(https://www.youtube.com/watch?v=Uj3_KqkI9Zo) Here's a Ted Talk about the thought experiment.

Misc. References:

"We must know. We will know"
This phrase can be found throughout the mission (in Hilbert's journal, on his chalkboard, and written in blood on the walls of the Aleph Stone exhibit room). It's an English translation of David Hilbert's epitaph - it just also seemed like the perfect thing for a madman consumed by a need to understand infinity to repeat, over and over.

Hilbert's chalk boards
There are four chalk boards in Hilbert's office. They depict:
1. (https://en.wikipedia.org/wiki/Hilbert%27s_problems) Hilbert's 23 problems, which were the 23 most important problems in mathematics, as posed by David Hilbert, in 1900. The first problem was the prove (or disprove) Cantor's "Continuum Hypothesis," which posits that there exists no set whose cardinality lies between that of the natural numbers and the real numbers.
2. The first 3 "Pseudo Hilbert Curves"
3. A proof of bijection between the real numbers and the interval [0,1]. This was an important first step in assessing the truth of Cantor's "Continuum Hypothesis".
4. Cantor's famous (https://en.wikipedia.org/wiki/Cantor%27s_diagonal_argument) "diagonal argument", which demonstrates the uncountability of the real numbers.
https://i.imgur.com/2Y2uPS6.png

https://i.imgur.com/OBNt46e.png

Extra-Dimensional Paintings
Throughout the mission, there are many "extra-dimensional paintings" which the player can enter. The inspiration comes the space-filling Hilbert Curve. As discussed above, the Hilbert Curve implies that a lower dimensional space can completely encode all information about a larger dimensional space. Thus, in principle, a 2-dimensional painting can encode all the information about a 3-dimensional space.

Cantor Set
One of Cantor's most famous inventions is the (https://en.wikipedia.org/wiki/Cantor_set) Cantor Set. The set is constructed from the interval [0,1]. First remove the middle third, then remove the middle third of the remaining 2 pieces, then remove the middle third of the remaining four pieces, etc...

This notion can be extended into 2-dimensions, yielding the (https://en.wikipedia.org/wiki/Sierpi%C5%84ski_carpet) Sierpinski Carpet, or into 3-dimensions, yielding the (https://en.wikipedia.org/wiki/Menger_sponge) Menger Cube.

Examples of these can be found throughout the mission. In the Gallery of Mathematical Curiosities, there is a triptych right outside the Aleph Stone exhbit room, which shows the Cantor Set, Sierpinski Carpet, and an interior view of the Menger Cube. In the next room, is the physical Menger Cube, which can be stolen as a bonus objective.

In Cantor's Temple, there are several Sierpinski Carpets. For example, the ceiling cutouts and floor pattern:

https://i.imgur.com/f0plXKf.png


Cantor's Paradise
Hilbert's journal refers to the place beyond the white stone gate in Cantor's Temple as "Cantor's Paradise." He further proclaims that "None shall stand between us and the paradise Cantor has created!"

This is based on the fact that Hilbert referred to set theory and infinite cardinal numbers (Cantor's inventions) as ( https://en.wikipedia.org/wiki/Cantor%27s_paradise) "Cantor's paradise," further declaring that: "No one shall expel us from the paradise which Cantor has created."


Hilbert Curves
The coat of arms symbol found on all the banners and guard uniforms is made up of the first three "pseudo-Hilbert curves." This shape also inspires the design of the Infinity Pool, as well as wood paneling in Hilbert and Brouwer's offices, the floor textures in the main hotel lobby and the stucco trim around much of the first floor:

https://i.imgur.com/fbJsN9j.png

Double helix spiral staircase
This is just a general homage is to notion of a space containing more than it seems it should be capable of holding. This is a theme in Hilbert's work, for example showing that a line segment 1 unit long contains exactly as many points as a 1x1 square.

The terms "Aleph" and "Omega"
These terms both come from Cantor's notation of "(https://en.wikipedia.org/wiki/Transfinite_number) transfinite numbers."

The term "Aleph" in "Aleph Stone" comes from Cantor's notation for "Transfinite Cardinals" (i.e., the sizes of infinite sets). "(https://en.wikipedia.org/wiki/Aleph_number#Aleph-naught) Aleph-naught" is the size of the smallest infinity (corresponding to the size of the set of the natural numbers), "(https://en.wikipedia.org/wiki/Aleph_number#Aleph-one) Aleph-one" is the size of the next smallest infinity (corresponding to the size of the set of the real numbers), etc...

The term "Omega" from "Omega Gallery" and "Omega Suites" comes from Cantor's notation for "transfinite ordinals." In Cantor's notation, after all infinity of the natural numbers comes Omega (then Omega+1, Omega+2, Omega+3, etc...). Since the hotel has infinitely many guest-room floors (numbered 1, 2, 3...), the floor right above them would naturally be the floor Omega (which contains the Suites and Gallery).

(https://www.youtube.com/watch?v=SrU9YDoXE88) Here's a silly but fun/informative video about transfinite numbers.

Supertask Elevator
A "(https://en.wikipedia.org/wiki/Supertask) Supertask" is a task that requires [countably] infinitely many steps, but can be finished in a finite amount of time. Hence why the elevators are "Supertask elevators"---the need to transport the player past infinitely many guest floors without taking forever.

Conceptually, it might seem ridiculous that infinitely many steps could be done in finite time. But consider a theoretical elevator that accelerates in the following manner:

The elevator requires one second to go from floor 1 to floor 2. Each floor the elevator ascends, its speed doubles. So it only takes 1/2 a second to go from floor 2 to floor 3, and only 1/4 seconds to go from floor 3 to floor 4, etc...

Such an elevator will ascend infinitely many floors in only 2 seconds, since the sequence 1+1/2+1/4+1/8+1/16+... converges to 2.

(https://www.youtube.com/watch?v=ffUnNaQTfZE) Here's a silly but fun/informative video about supertasks.

Brouwer
Brouwer was a mathematical contemporary of Hilbert's. Hilbert considered him to be both brilliant and feeble-minded (in large part due to their differences of opinion regarding ‘formalism' and ‘intuitionism').

Brouwer is most famous for his (https://en.wikipedia.org/wiki/Brouwer_fixed-point_theorem) fixed point theorem. The theorem is represented on the chalkboard in his office, and is the inspiration behind his painting:

https://i.imgur.com/PeZUMcc.png

(https://www.youtube.com/watch?v=csInNn6pfT4) Here's a silly but fun/informative video about fixed points


Keeper Zermelo
Zermelo was one of Hilbert's students.

downwinder on 7/9/2020 at 17:35

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i would love to see a mission with time travel

Samantha1 on 11/9/2020 at 18:45

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I´ve just replayed your great mission and read this post. I gotta say its clear how much effort you put into this. Clever & astonishing. Great job :thumb: :)