"… the Jews have discovered a way to access a fourth spatial dimension."
— Clifford Pickover, description of his novel Jews in Hyperspace
"If you have built castles in the air, your work need not be lost;
that is where they should be. Now put the foundations under them.”
— Henry David Thoreau
"King Solomon's Mines," 1937—
The image above is an illustration from "Romancing the Hyperspace," May 4, 2010.
Happy birthday to the late Salomon Bochner.
260813-Palm_Swings-page-at-IMDb-displayed-banner-ad-
for-Disney_Worldbuilders-'Coming-Aug-20'.jpg
This banner ad appeared tonight above the IMDb "Palm Swings" trailer.
"If you have built castles in the air, your work need not be lost;
that is where they should be. Now put the foundations under them.”
— Henry David Thoreau
A space where author Madeleine L'Engle worked as a librarian —
A more abstract space —
"If you have built castles in the air, your work need not be lost;
that is where they should be. Now put the foundations under them.”
— Henry David Thoreau
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From a Google AI Mode search (updated version of June 12, 2026) —
You said: [ Using an AI-Mode-suggested prompt ]
Map out how three-premise Aristotelian syllogisms (like Barbara: All A is B, All B is C…) travel through a higher-dimensional space.
June 11, 2026
When moving from two propositions ((P) and (Q)) to three classes ((A), (B), and (C)), the geometric logic space undergoes a massive expansion. Instead of a 4-dimensional space with 16 tiles, we enter an 8-dimensional hypercube (an octeract) containing (28 = 256) logical operations [1].
In this higher-dimensional space, categorical syllogisms like Barbara (All (A) is (B), All (B) is (C), therefore All (A) is (C)) manifest as an invariant, directional path across specialized Venn-like sub-planes of the 8D cube. [1]
1. Constructing the 8-Dimensional Coordinate
For three terms, a Venn diagram has exactly (23 = 8) distinct spatial regions. To track the space, we assign an 8-bit coordinate (𝑏7, 𝑏6, … , 𝑏0) where each bit acts as a switch representing whether a specific region is populated (1) or empty (0).
The standard order for these 8 partitions based on set combinations is:
Regions = [ABC, AB¬ C, A¬ BC, A¬ B¬ C, ¬ ABC,¬ AB¬ C, ¬ A¬ BC, ¬ A¬ B¬ C]
In Aristotelian logic, a statement like "All (A) is (B)" means the region containing elements that are (A) but not (B) is completely empty (0). The remaining regions remain possible (1).
2. Translating the Premises into 8D Vectors
Let's map out the exact coordinates for the premises of the Barbara syllogism:
3. The 8D Intersection (The Starting Vertex)
To evaluate the syllogism, we find the logical conjunction (∧) of both premises. Geometrically, this means computing the bitwise intersection of our two 8D hypercube vertices:
VPremises = (1,1,0,0,1,1,1,1) · (1,0,1,0,1,0,1,1) = (1,0,0,0,1,0,1,1))
This intersection lands us on a highly specific Weight-4 vertex sitting exactly on the equator of the 8-dimensional hypercube. This point represents the combined structural reality constraints of our premises.
4. Slicing Upward to the Conclusion
A syllogism is valid if its conclusion is a geometric ancestor of this starting vertex in the 8D Inclusion Lattice. Let's look at the coordinate for the target conclusion:
Now we map the trajectory from our premise vertex to our conclusion vertex:
Because every single
1 in our premise vector is perfectly preserved in the conclusion vector, the geometry establishes a definitive upward pathway from Weight-4 to Weight-6. The conclusion opens up two extra dimensions of freedom (slots 3 and 6 switch from (0) to (1)), proving that the constraints of the premises inherently contain and imply the conclusion.
Conclusion
The Barbara syllogism operates as an active path-finding engine inside an 8-dimensional space. By intersecting the (0)-planes (the empty spaces) of the premises, the logic forces a particle to start at a constrained 8D vertex. Because the target conclusion sits directly upward on the lattice, walking the path of deductive reasoning is structurally identical to climbing along the directional edges of an 8-dimensional octeract.
If you want to continue navigating higher-dimensional logical shapes, let me know if you would like to:
[ Reference favicons for 3 cited sites ]
  
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Exercise . . .
Check the accuracy of the above AI Mode statements.
Related reading . . .
A search in this journal for "Many Dimensions"
"If you have built castles in the air,
your work need not be lost;
that is where they should be.
Now put the foundations under them.”
— Henry David Thoreau
Related documentation . . .
The foundation for the above documentation is shown
by Google in the AI Overview and search result below.
See as well this journal on the above YouTube date — Dec. 4, 2009.
"If you have built castles in the air,
your work need not be lost;
that is where they should be.
Now put the foundations under them.”
— Henry David Thoreau
"When you build your house
Then call me home"
— Fleetwood Mac, "Sara"
“If you have built castles in the air,
your work need not be lost;
that is where they should be.
Now put the foundations under them.”
— Henry David Thoreau
The new URL matrix.bingo forwards to
http://m759.net/wordpress/?s=5×5 .
“If you have built castles in the air,
your work need not be lost;
that is where they should be.
Now put the foundations under them.”
— Henry David Thoreau
"If you have built castles in the air,
your work need not be lost;
that is where they should be.
Now put the foundations under them.”
— Henry David Thoreau
See as well 5×5, The Matrix of Abraham, and Deutsche Schule Montevideo .
“If you have built castles in the air,
your work need not be lost;
that is where they should be.
Now put the foundations under them.”
— Henry David Thoreau
“If you have built castles in the air, your work need not be lost;
that is where they should be. Now put the foundations under them.”
— Henry David Thoreau
Between, or: Interality Illustrated (according to Sherald)
“We live in between the lines;
in between the cracks . . . .
That’s where the bonds are formed. ”
"… which grounds the self" . . .
Popular Mechanics online today —
"Verizon exec Marni Walden seemed to
indicate Mayer's future may still be up in the air."
See also 5×5 in this journal —
"If you have built castles in the air,
your work need not be lost;
that is where they should be.
Now put the foundations under them.”
— Henry David Thoreau
From yesterday's 2 PM post —
From "Inception" —
Paraphrase of remarks by "Inception" director Christopher Nolan
at Princeton on June 1, 2015 —
"If you have built castles in the air,
your work need not be lost;
that is where they should be.
Now put the foundations under them.”
— Henry David Thoreau
Princeton University's president on June 2, 2015 —
“Dream Audaciously,” Eisgruber Urges Graduates
Related news —
Film director Christopher Nolan at Princeton on June 1, 2015:
“In these graduation speeches, generally, you have the
speaker say something along the lines of, ‘You need to
chase your dreams,’ ” Nolan said. “But I’m not going to
say that because I don’t believe it. I don’t want you to
chase your dreams. I want you to chase your realities.
And I want to say: Don’t chase your realities at the
expense of your dreams, but as the foundation of your
dreams.”
"If you have built castles in the air,
your work need not be lost;
that is where they should be.
Now put the foundations under them.”
— Henry David Thoreau
"A vast space that travels down to the bedrock
upon which the towers were built, the museum
winds its way deeper and deeper underground,
taking visitors on a journey to the very bottom."
— The Associated Press in
this evening's Washington Post
This suggests a review of a different sort of
bedrock:—
"If you have built castles in the air,
your work need not be lost;
that is where they should be.
Now put the foundations under them.”
— Henry David Thoreau
» more
This, together with this morning's post involving the squares 16 and 9,
suggests a review of Conceit at Harvard (October 25, 2006), which
contains the following figure involving the squares 16, 9, and 25—
"If you have built castles in the air,
your work need not be lost;
that is where they should be.
Now put the foundations under them.”
— Henry David Thoreau
"Orthodox Jews are disappearing from Jerusalem. One moment they are praying at the Western Wall, and in the blink of an eye, they seem to evaporate…. In order to build the Third Temple while being respectful of the Islamic structures on the Temple Mount, the Jews have discovered a way to access a fourth spatial dimension. They will build the Third Temple invisibly 'above' the Temple Mount and 'above' the Mosque in the direction of the fourth dimension."
— Clifford Pickover, description of his novel Jews in Hyperspace
"If you have built castles in the air, your work need not be lost; that is where they should be. Now put the foundations under them."
— Henry David Thoreau, conclusion of Walden
Related material: Log24 entries, morning and evening of June 11, 2009, "Text" (June 22, 2009), and Salomon Bochner's remarks on space in "Eight is a Gate" (Feb. 26, 2008).
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