Imagine, if you will, the ultimate life hack
for the modern housing market. You know, you
want more square footage, but physical land is
at a premium. Right. Especially in a city. Exactly.
So what if you could just bypass the limitations
of geography entirely? What if you could save
an absolute fortune on real estate simply by
building a house that folds out into the fourth
dimension? I mean, it sounds like the setup to,
well, a highly specialized, very nerdy architectural
joke. It does. It sounds completely absurd, but
that is exactly the bizarre proposition we are
exploring today. So welcome to the deep dive.
We are thrilled to have you with us because today's
source material is an absolute classic of science
fiction, and it actually doubles as a master
class in theoretical mathematics. Yeah, it really
does. We are looking at a Wikipedia article detailing
a story by the legendary American writer Robert
A. Heinlein. The story is called And He Built
a Crooked House, which, of course, is a play
on the old nursery rhyme. you know, there was
a crooked man. Right, and it was published all
the way back in February of 1941. Wow, 1941.
Yeah, an astounding science fiction magazine.
It is this amazing piece of literature that has
been baffling and educating readers for over
80 years now. And our mission today is to explore
how this vintage sci -fi story about a mathematically
obsessed, highly eccentric architect somehow
serves as the perfect vehicle for understanding
four -dimensional geometry. Okay, let's unpack
this. Well, to understand the genius of this
story, we really must start with the literal
foundation. We are introduced to our protagonist,
Quintus Teal. Right. He is a graduate architect
operating in the Los Angeles area. And this is
during a time when the city was experiencing
a massive boon in experimental architecture.
Right, a lot of weird houses being built. Exactly.
But Thiel is not interested in designing, you
know, standard Spanish colonials or mid -century
moderns. He is deeply, deeply obsessed with topology
and something called the Picard Vessio theory.
Okay, for those of you listening who might not
have a degree in advanced mathematics, let's
break that down a bit. Topology is often described
using a classic analogy. Oh, the coffee mug one.
Yeah, exactly. So to a topologist, a coffee mug
and a donut are the exact same shape. Which sounds
insane to anyone else. Right. But because topology
studies the properties of a geometric object
that are preserved under continuous deformations
like stretching or bending, but crucially not
tearing. So since both a donut and a coffee mug
have exactly one hole, they are topologically
identical. It's the perfect way to visualize
it. I mean, a topologist doesn't care about rigid
distance. or angles, you know, they care about
connectivity. And then you add in his obsession
with the Picard -Vizio theory, which is, well
essentially it's the Galois theory of differential
equations. Okay, now you're losing me again.
Basically it deals with finding the hidden symmetries
for complex mathematical systems. So Thiel wants
to take these highly abstract concepts, symmetries
of equations and the fluid connectivity of shapes,
and actually apply them to physical lumber and
plaster. Which creates a beautiful, yet kind
of terrifying friction. Because, I mean, in architecture,
rigid distances and angles are literally the
only things keeping a roof over your head. But
Thiel takes this obsession to his friend Homer
Bailey, and he pitches a house design that is
just mind -boggling. He wants to build a house
shaped like the unfolded net of a tesseract.
Right. A tesseract being the four -dimensional
equivalent of a cube. Just as a flat two -dimensional
square can be extruded into a three -dimensional
cube, a cube can be mathematically extended into
a four -dimensional tesseract. Let me try to
visualize this with an analogy for you. Imagine
you take a regular, everyday cardboard box. If
you take a box cutter and you slice along the
vertical edges and lay it completely flat on
the ground, it forms the shape of a cross. You
have just taken a 3D object and unfolded it into
a 2D plane. Teal is essentially doing that exact
same process, but taking it one dimensional level
higher. He wants to unfold a 4D object into our
3D world. He even uses toothpicks and clay to
map out the projections for his friend just to
prove it can actually be done. And the structure
they ultimately design and build is described
as an inverted double cross. It consists of eight
cubical rooms in total. So there is a central
vertical stack of four cubes. Then surrounding
the second cube from the bottom, there are four
additional cubes jutting out in each cardinal
direction. Like a cross. Yeah, this is the classic
mathematically accurate 3D footprint of a 4D
shape. But wait, I have to push back on this
from a pure engineering standpoint. You have
a central tower of four rooms and then four separate
rooms just hanging off the sides of the second
floor. Suspended in midair, basically. Yeah.
Practically speaking, how does an inverted double
cross of cubical rooms not just immediately tip
over under its own weight like it sounds structurally
impossible? What's fascinating here is that Heinlein
entirely bypasses the deep engineering logistics
by leaning into Thiel's eccentricities and his
surprisingly grounded motivation. Wait, grounded?
Yeah, I mean, Thiel isn't just some mad scientist
trying to break physics for the fun of it. His
motivation is highly humorous and practical.
It's California real estate costs. Oh, of course.
Right. By building this structure in this specific
footprint, Teal is getting an eight -room house
while only taking up the ground space and, crucially,
paying for the land of a single cubicle room.
That is, I mean, it's the ultimate density workaround.
He is gaming the Los Angeles zoning laws using
theoretical topology. Exactly. So they actually
build this magnificent, impossible -looking 3D
structure. The foundation is laid, the precarious
rooms are stacked, but... You know, building
a structurally dubious house in Southern California
comes with a very specific geographic ticking
clock. The fault lines. The San Andreas fault,
to be exact. Right. So the night before Teal
is scheduled to give his friend Homer and Homer's
wife Matilda a grand tour of the finished house,
a classic California earthquake strikes the region.
A big one. And the next morning, when Teal, Homer
and Matilda arrive at the property, the grand
eight room inverted double cross is gone. Just
vanished. Or at least it appears to be. From
the outside, the house just looks like a single
cubicle room sitting on the lot. The entire upper
structure and those four cubes hanging off the
sides have seemingly vanished into thin air.
Right. So naturally, they go inside to see what
happened, and inside the house, the laws of physics
are entirely broken. Yeah, the spatial relationships
between the rooms have gone completely mad. They
find that all the upper floors are completely
intact. Which is what? The eight rooms are still
there? but the stairs now form a dizzying closed
loop. Okay, I want you to picture this in your
own home. Imagine walking up the stairs from
your ground floor, reaching the second floor
landing, deciding to take another flight up to
the attic, and as you reach the top of those
stairs, you step right back into your ground
floor living room. It's totally paradoxical.
You walk up and somehow you end up exactly below
where you started. And it isn't just the stairs.
All the doors and windows inside the house lead
directly into other interior spaces. There is
no exterior anymore. You open a front door expecting
to look outside at the street and you are just
looking into another hallway. They are entirely
sealed off from the outside world. Here's where
it gets really interesting. At one point, the
trio looks down a long hallway and they are shocked
to see their own backs. Yes. I cannot stop thinking
about the sheer visual horror of that moment.
Imagine standing in a hallway looking forward
and seeing the back of your own head staring
down the same hallway. But from a physics perspective,
how does that work? Light travels in straight
lines. How is a reader, or anyone for that matter,
supposed to mentally map a space where a straight
line of sight circles back on itself? Well, to
understand that, we have to look at what the
earthquake actually did. Okay. The earthquake
provided this necessary cosmic jolt. It wasn't
just shaking the ground. It essentially acted
as the box cutter folding the cardboard box back
together. Oh, wow. Yeah. It folded the 3D net
of the house into an actual fully functional
4D tesseract. So the rooms haven't moved in a
way we can physically perceive, but their boundaries
now intersect through the fourth dimension. Exactly
that. Just as folding a 2D cardboard cross creates
a 3D box where the flat edges suddenly touch
in new ways, the earthquake folded their 3D rooms
into a higher dimension. And the light bending.
Right. Regarding the light. Light does travel
in a straight line through space. But if space
itself is curved and folded through a fourth
dimension, that straight line wraps around. So
like walking on a globe. Exactly. Much like how
an airplane flying in a quote unquote straight
line over the Earth's surface eventually circles
the globe. When they look down the hallway and
see their own backs, their line of sight is traveling
through the curved space of a tesseract and wrapping
right back to their own retinas. That is just
wow. So they are entirely enclosed in a shape
that has no outside, at least from their three
-dimensional perspective. They are trapped in
a loop where every single door leads to another
hallway. By nightmare. Realizing they are essentially
rats in a geometric maze, they decide they only
have one option left to try and interact with
the physical world, and that's the windows. And
this is where the house shifts from being a mere
architectural puzzle to quite literally a window
into the multiverse. Teal tries the first seemingly
logical exit. He opens a French window on the
ground floor, fully expecting to step into the
dining room, but instead... He falls awkwardly
outside and lands face first in some shrubbery.
Not very dignified. No, it is a very ungraceful
exit for an architect of his stature. But it
proves the house still has some tenuous connection
to the real world. However, the real chaos happens
when they explore the original top room of the
house. Right, because the windows in this room
no longer connect to mathematically logical spaces.
They act like portals. Let me walk you through
what they see out of these four windows, because
the imagery is just wild. It really is. Window
number one gives them a dizzying, vertigo -inducing
view from high above the Empire State Building
in New York. Across the entire country. Right.
Then window number two reveals an upside -down
view of a seascape, like the ocean is literally
hanging above them. And then window number three
is the most unsettling. It looks out into a place
of absolute no space. The story describes it
as having no color at all, not even black. just
complete, incomprehensible nothingness. Yeah,
that one gives me the chills. And finally, window
number four looks out onto an unearthly, bizarre
desert scene. This raises an important question
about how we perceive our reality and how Heinlein
uses these completely disconnected windows to
illustrate a profound mathematical point. A higher
dimension completely mocks our standard 3D understanding
of geography. Because in 4D space, two points
that are thousands of miles apart in 3D space
might actually be adjacent. Think about a piece
of paper. If you draw two dots on opposite ends
of a long sheet of paper, they are far apart
in two dimensions. Sure. But if you take that
paper and fold it through the third dimension
so the two dots touch, the distance between them
becomes zero. Oh, I see. The house, now existing
in four dimensions, is doing exactly that to
our universe. It is intersecting different points
of our 3D world simultaneously. So the Empire
State Building, an upside -down ocean, and an
unknown desert are all just next door from the
perspective of a tesseract's adjacent faces.
That perfectly explains the geographic jumps.
But what about that third window, the no space?
You know, it isn't described as black, but as
an absence of everything. Well, that is Heinlein
attempting to describe the sensory deprivation
of looking outside the bounds of our universe
entirely. Outside the universe. Yeah. If the
house is a 4D object intersecting our 3D plane,
some of its faces might not intersect with our
universe at all. So looking out that window is
looking into the void between dimensions. There
is no light to be black. It is the visual equivalent
of a blind spot. That is so conceptually heavy.
And while they're trying to process this impossible
panorama and the sheer existential dread, of
the no space, a second earthquake hits. The house
starts shaking violently in an absolute panic,
realizing the geometry might shift into an even
worse configuration or trap them in that void
forever. They all dive head first through the
desert window. It is a total leap of faith. And
they land amid twisted, bizarre, tree -like vegetation.
And the window they just jump through completely
vanishes behind them. Right. So at first, looking
at these weird trees, they are terrified that
they have been transported to an alien planet.
But then comes the twist. A passing truck driver
spots them and casually informs them that they
are simply in Joshua Tree National Park. I love
that. Or, as the driver refers to it, Joshua
Tree National Forest, which is just a couple
of hours east of Los Angeles. So the desert window
wasn't an alien world at all. It was just a fold
in the map of Southern California. I have to
point out the sheer absurdity of their reactions
here though, especially Thiel's. I mean, he built
a house that shatters the laws of reality, traps
him in an infinite loop, shows him the absolute
void of no space, and his response throughout
the ordeal is to just excitedly go exploring.
Like, is the man entirely sane? Well, Thiel's
sanity isn't the issue. He's just adapting to
the math way faster than his friends are. His
obsession with the Picard -Vessio theory means
he already accepts that the universe operates
on unseen symmetries. To him, the house is just
confirming his life's work. I guess that makes
an eerie kind of sense. So our characters are
safe among the Joshua trees, but what happens
to the house itself? After finding their way
back from the desert, they return to the original
Hollywood construction site, but the house has
completely vanished. There's not a single toothpick
or brick left. Teal, ever the unfazed architect,
nonchalantly deduces that during the second earthquake,
the house must have fell through into another
section of space. And falling through space is
actually a brilliant way to describe a 4D movement.
How so? Imagine a two -dimensional world, like
a flat piece of paper. If a 3D sphere passes
through that paper, the 2D inhabitants would
just see a circle appear, grow larger, shrink,
and then completely vanish. Oh, because they
only see the slice of the sphere intersecting
their plane. Exactly. The sphere didn't cease
to exist. It just moved along a spatial axis
the 2D world cannot perceive. The house did the
exact same thing. It shifted slightly along the
fourth -dimensional axis, effectively vanishing
from our 3D coordinate system. And Teal's only
regret. He casually notes that he really should
have anchored it at the foundations, as if a
few more steel bolts would have stopped a four
-dimensional spatial fold. Right. The story ends
wonderfully. Teal is already excitedly talking
about a great new revolutionary idea for a house.
Homer Bailey, who has been put through absolute
dimensional hell, finally snaps and tries to
punch Teal right in the face. But Teal effortlessly
dodges the punch because, as the story notes,
he was always a man of action. It is a comedic,
perfectly paced ending to a genuinely mind bending
premise. And there are a few incredible real
world details attached to this story that really
anchor it to history. The actual given for Quintus
Teal's architectural nightmare is 8775 Lookout
Mountain Avenue in Hollywood. Wait, really? Yeah.
And that location is described in the story as
being across the street from the hermit, the
original hermit of Hollywood. What is amazing
is that 8775 Lookout Mountain Avenue was actually
across the street from Robert A. Heinlein's real
-life home at the time he wrote the story. That's
hilarious. He essentially placed this... reality
breaking, world folding house right in his own
front yard. It's a great Easter egg and beyond
the fun trivia. This story had a massive cultural
and scientific impact. In a 1978 article for
the New York Times, the legendary astronomer
Carl Sagan explicitly praised, and he built a
crooked house. Wow, Carl Sagan. Yeah. Sagan stated
that for many readers, this story was the first
introduction to four -dimensional geometry that
held any promise of comprehensibility. That quote
from Sagan is so telling. It really highlights
how fiction acts as a Trojan horse for hard science.
If you hand the average person a dense textbook
on geometric topology, they will likely close
it in five minutes. The abstract math is simply
too disconnected from human experience. But if
you hide that exact same math inside a witty
survival story about a greedy architect trapping
his friends in a haunted geometry house, they
are suddenly learning complex physics without
even realizing it. If we connect this to the
bigger picture, Sagan is hitting on a fundamental
truth about human cognition. We are biologically
wired for narrative. Yeah, we remember stories.
Exactly. We remember cause and effect, struggle
and resolution far better than we remember raw
data. Heinlein took an abstract spatial concept
and turned it into a sensory experience. You
feel the vertigo of the Empire State window,
you feel the claustrophobia of the looping stairs,
and we see the ripple effects of this narrative
approach in so many other works. Right, the Wikipedia
article actually notes several spiritual successors
and thematic cousins to Heinlein's story. Flatland,
for example, the classic Victorian satire, uses
the perspective of a two -dimensional square
to force readers to comprehend the limitations
of their own dimensions. And then you have popular
television, Star Trek. Voyager features an episode
called Twisted, where the ship encounters an
anomaly that subjects the interior to mysterious,
seemingly impossible physical reconfigurations.
Like the Cricket House. Exactly. The corridor
stopped making sense, and the crew cannot reach
the bridge, the classic tesseract problem. There
is also a 1950 short story mentioned called A
Subway Named Mobius, where the Boston subway
system accidentally acquires an extra -dimensional
topology, causing a train to vanish into a mathematical
fold. All of these stories take incredibly complex
geometry and turn it into a comprehensible, gripping
human problem. If you have ever sat in a classroom
or been in a meeting feeling entirely overwhelmed
while trying to learn a highly abstract concept,
this is exactly why narrative matters. It isn't
just entertainment. It is the ultimate tool for
turning the inaccessible into the unforgettable.
Beautifully said. When you attach a dry fact
to a human struggle, even a funny one like an
architect dodging a punch from his frustrated
friend, it sticks in your brain forever. So what
does this all mean? It means the next time you
are trying to understand something difficult,
try to find the story in it. Look for the quintis
teal in the math. And if you find yourself house
hunting in Los Angeles, maybe have a topologist
look over the blueprints before you sign anything.
Excellent advice. I want to leave you with one
final thought today. Next time you are sitting
in a perfectly square, normal room, well, take
a moment to really look at the walls, the ceiling,
and the floor around you. We assume these rigid
right angles are the absolute truth of our universe.
But what if our entire three -dimensional reality,
everything we know and can touch, is just an
unfolded blueprint of a much larger, unseen structure?
And what if it's just waiting for the right kind
of earthquake to finally snap it into its true
shape? Until next time, keep diving deep.