A Crooked House

SUBHEAD: Revisiting a 1974 architectural theses project that laid out three Tiny Homes in a 8' wide 4D cube.

By Juan Wilson on 11 December 2019 for Island Breath -
(https://islandbreath.blogspot.com/2019/12/a-crooked-house.html)


Image above: Plan view of 8'x8' of Tiny Home and 8'x8' Tiny Yard. Home features single bed, kitchen counter with sink, work desk, shelving, cabinets, composting toilet with pass-thru panels connecting to other two homes. Click to enlarge.

Note: This is the initial posting of this work on our blog site and it will likely be upgraded and re-edited before it is finalized.

In the 1950's, when I was a teenager, I read short science-fiction story titled 'And He Built a Crooked House'. It is a science fiction short story by American writer Robert A. Heinlein, first published in Astounding Science Fiction in February 1941. It was reprinted in the anthology Fantasia Mathematica in 1958 and in the Heinlein collection The Unpleasant Profession of Jonathan Hoag in 1959.

The story is about a mathematically inclined architect named Quintus Teal who has what he thinks is a brilliant idea to save on real estate costs by building a house shaped like the unfolded net of a tesseract. The title is paraphrased from the nursery rhyme "There Was a Crooked Man". See (https://en.wikipedia.org/wiki/%22%E2%80%94And_He_Built_a_Crooked_House%E2%80%94%22).

The form of this unfolded 4D cube (or tesseract) took was a four story house of stacked cubes with four additional cubes wrapped around an upper floor. 

At the time I had some interest in architecture and was fascinated by Heinlein's story. More than 20 year, while studying architecture at the Cooper Union in New York City I became interested in the representation of geometric objects through multidimensional space.

In 1973-74, as part of my Fifth Year Theses Project I investigated representing various 4D object concepts like "cube", "tube", "cup", "bottle" in drawing and model formats. One of my studies was a reevaluation of Heinlein's Crooked House architecture.

I decided to design a 4D regular cube that would be 8feet x 8feet x 8feet x 8feet. This hypercube would have there for have 8 8ft x 8ft x 8ft 3D volumes attached to one another as the "faces" of the 4D cube.

The eight 3D cube spaces include:

 - One 8' cube of Earth with a small hyper-dense central rock cor providing One G of gravity to adjacent spaces. The earth cube is living soil providing needed organic materials and minerals to support plants.

-  Three 8' cube Yards with grass and small tree rooted in the Earth cube. There is also a Welcome mat. The volume of the cube filled with breathable air.

 - Three 8' cube Tiny Homes with a door (each facing a Welcome mat in one of the yards). Units also feature a wall with a window (facing an adjacent yard) a ceiling with skylight with access to the roof. There are two adjacent walls in each unit that abut the other two units.These otherwise blank walls each have a 16"x16" pass-through door that allow exchanging items between the other Tiny Home units without going outside. One pass-through door is red and the other green.

- One 8' cube of Sky A space that is above the roofs of the three living units and above the yards. At the center of the sky cube is a small sun-like sphere that provide light and energy for units and yards. At the center of the sphere is a small black hole 16" in diameter that is the only way in and out of this small universe.


Image above: Isometric views of 8'x8' elements, or parts, of 4D Tiny Home project include Home, Yard, Overhead Sky and Underfoot Earth. 

The three Yard and Home cubes are arranged so that the six faces of the 8' Earth Cube supports the bottom of three Yards and three Tiny Homes.

It should be noted that the eight face cube volumes of a 4D hypercube are merely faces... much like the six faces of a empty cardboard box. It is usually what's inside the 3D space of the cardboard box that is the real prize.

Similarly the eight cube faces are merely the surface of the 4D cube. What is inside the 4D cube that is the real content... In this case maybe something to keep the Earth Cube fertile and the Yard Cubes breathable and moist and the Sky cube protective and a source of energy.

When the eight volumes of the tesseract are folded into the four dimensional hypercube the three Tiny Homes are abutted to one another and the three yards are contiguous.


Image above: Isometric views of eight 8'x8'x8' elements of 4 dimensional Tiny Home project arranged into tesseract (4D unfolded cube analogous to a unfolded cardboard box.

The three Yard and Home cubes are arranged so that the six faces of the 8' Earth Cube supports the bottom of three Yards and three Tiny Homes.

It should be noted that the eight face cube volumes of a 4D hypercube are merely faces... much like the six faces of a empty cardboard box. It is usually what's inside the 3D space of the cardboard box that is the real prize.

Similarly the eight cube faces are merely the surface of the 4D cube. What is inside the 4D cube that is the real content... In this case maybe something to keep the Earth Cube fertile and the Yard Cubes breathable and moist and the Sky cube protective and a source of energy.




Image above: Two views, each showing four "sides" of the Hypecrcube's eight volumes when they are folded into the fourth dimension. 

When the eight volumes of the tesseract are folded into the four dimensional hypercube the three Tiny Homes are abutted to one another and the three yards are contiguous. The two views above are analogous to looking at the opposite sides of a 3D dice, where you can only see the number of spots on 3 of the 6 sides of the dice. In the case of a 3D dice there are 8 unique views of 3 sides of the dice. In the case of a 4D cube there are 16 views of any four corner adjacent of the 8 volumes.



Image above:  Pencil drawing from 5th year architectural thesis year at the Cooper Union by Juan Wilson in 1974. Isometric view of unfolded 4D Hypercube showing four of its eight volumes.


Image above: Pencil drawing from 5th year architectural thesis year at the Cooper Union by Juan Wilson in 1974. Isometric view from one side of the Hypercube showing four of its eight volumes. 


Image above: Salvador Dali's 4D crucifixion titled "Corpus Hypercubus" from 1954. In his 1951 essay "Mystical Manifesto",Dali introduced an art theory he called "nuclear mysticism" that combined his interests in Catholicism, mathematics, science, and Catalan culture in an effort to reestablish classical values and techniques, which he extensively utilized in Corpus Hypercubus. From (https://en.wikipedia.org/wiki/Crucifixion_(Corpus_Hypercubus)).

 

Video above: Animations of rotating four dimensional cubes, tetrahedrons and Spheres by Eugene Khutoryansky. From (https://www.youtube.com/watch?v=eyuRLmCphHc).

 

.

Iranian woman mathematician dies

SUBHEAD: Maryam Mirzakhani, first woman to win the Fields Medal - the "Nobel Prize for Mathematics"

By Staff on 15 July 2017 for the BBC News -
(http://www.bbc.com/news/science-environment-40617094)


Image above: Stanford faculty photograph of Maryam Mirzakhani. From (https://mathematics.stanford.edu/people/name/maryam-mirzakhani/).

[IB Publisher's note: My daughter Laura was born in Tehran, Iran in 1976, just about the time Maryam was born. I was working there on a one year contract for a planning firm. I found the Iranians to be wonderful people. When I finished the following article I felt stricken as if she were my daughter too. It is incomprehensible to me that we perceive the Iranians to be evil while we buddy up to the Saudis royals as our friends.]

Nearly three years after she became the first woman to win math's equivalent of a Nobel Prize, Maryam Mirzakhani has died of breast cancer at age 40. Her death was confirmed Saturday by Stanford University, where Mirzakhani had been a professor since 2008.

Mirzakhani is survived by her husband, Jan Vondrák, and a daughter, Anahita — who once referred to her mother's work as "painting" because of the doodles and drawings that marked her process of working on proofs and problems, according to an obituary released by Stanford.

"A light was turned off today .... far too soon. Breaks my heart," former NASA scientist Firouz Naderi said in a tweet.

He later added, "A genius? Yes. But also a daughter, a mother and a wife."

Naderi later posted a time-lapse video of Mirzakhani presiding over a lecture hall, filling chalkboards with a proof.

Early in her life, Mirzakhani had wanted to be a writer. But her passion and gift for mathematics eventually won out.

"It is fun — it's like solving a puzzle or connecting the dots in a detective case," Mirzakhani said when she won the prestigious Fields Medal in 2014. "I felt that this was something I could do, and I wanted to pursue this path."

Mirzakhani was born in Tehran, Iran, and she lived in that country before coming to the U.S. to attend graduate school at Harvard University. By then, she was already a star, having won gold medals in the International Mathematical Olympiad in the mid-1990s — after becoming the first girl ever named to Iran's team.

"There were more accolades," Danielle Karson reports for NPR's Newscast unit. "Mirzakhani was the first Iranian woman elected to the National Academy of Sciences last year, in recognition of her 'distinguished achievement in original research.' She was in good company: Albert Einstein, Thomas Edison and Alexander Graham Bell were past honorees."

Describing Mirzakhani's work, Stanford says:

"Mirzakhani specialized in theoretical mathematics that read like a foreign language by those outside of mathematics: moduli spaces, Teichmüller theory, hyperbolic geometry, Ergodic theory and symplectic geometry.

"In short, Mirzakhani was fascinated by the geometric and dynamic complexities of curved surfaces — spheres, doughnut shapes and even amoebas. Despite the highly theoretical nature of her work, it has implications in physics, quantum mechanics and other disciplines outside of math. She was ambitious, resolute and fearless in the face of problems others would not, or could not, tackle."

.

Time and Age

SUBHEAD: Reasons why time seems to go by more quickly as we get older.

By Christian Yates on 12 August 2016  in AlterNet  -
(http://www.alternet.org/personal-health/why-time-seems-go-more-quickly-we-get-older)


Image above: Detail of illustration of Father Time. From (http://mysticinvestigations.com/supernatural/father-time-watch/).

When we were children, the summer holidays seemed to last forever, and the wait between Christmases felt like an eternity. So why is that when we get older, the time just seems to zip by, with weeks, months and entire seasons disappearing from a blurred calendar at dizzying speed?

This apparently accelerated time travel is not a result of filling our adult lives with grown-up responsibilities and worries. Research does in fact seem to show that perceived time moves more quickly for older peoplemaking our lives feel busy and rushed.

There are several theories which attempt to explain why our perception of time speeds up as we get older. One idea is a gradual alteration of our internal biological clocks. The slowing of our metabolism as we get older matches the slowing of our heartbeat and our breathing. Children’s biological pacemakers beat more quickly, meaning that they experience more biological markers (heartbeats, breaths) in a fixed period of time, making it feel like more time has passed.

Another theory suggests that the passage of time we perceive is related to the amount of new perceptual information we absorb. With lots of new stimuli our brains take longer to process the information so that the period of time feels longer.

This would help to explain the “slow motion perception” often reported in the moments before an accident. The unfamiliar circumstances mean there is so much new information to take in.

In fact, it may be that when faced with new situations our brains record more richly detailed memories, so that it is our recollection of the event that appears slower rather than the event itself. This has been shown to be the case experimentally for subjects experiencing free fall.

But how does this explain the continuing shortening of perceived time as we age? The theory goes that the older we get, the more familiar we become with our surroundings. We don’t notice the detailed environments of our homes and workplaces.

For children, however, the world is an often unfamiliar place filled with new experiences to engage with. This means children must dedicate significantly more brain power re-configuring their mental ideas of the outside world. The theory suggests that this appears to make time run more slowly for children than for adults stuck in a routine.


So the more familiar we become with the day-to-day experiences of life, the faster time seems to run, and generally, this familiarity increases with age.

The biochemical mechanism behind this theory has been suggested to be the release of the neurotransmitter dopamine upon the perception of novel stimuli helping us to learn to measure time. Beyond the age of 20 and continuing into old age, dopamine levels drop making time appear to run faster.

But neither of these theories seem to tie in precisely with the almost mathematical and continual rate of acceleration of time.

The apparent reduction of the length of a fixed period as we age suggests a “logarithmic scale” to time. Logarithmic scales are used instead of traditional linear scales when measuring earthquakes or sound. Because the quantities we measure can vary to such huge degrees, we need a wider ranging measurement scale to really make sense of what is happening. The same is true of time.

On the logarithmic Richter Scale (for earthquakes) an increase from a magnitude ten to 11 doesn’t correspond to an increase in ground movement of 10% as it would do in a linear scale. Each increment on the Richter scale corresponds to a ten-fold increase in movement.

Toddler time
But why should our perception of time also follow a logarithmic scaling? The idea is that we perceive a period of time as the proportion of time we have already lived through. To a two-year-old, a year is half of their life, which is why it seems such an extraordinary long period of time to wait between birthdays when you are young.

To a ten-year-old, a year is only 10% of their life, (making for a slightly more tolerable wait), and to a 20-year-old it is only 5%. On the logarithmic scale, for a 20-year-old to experience the same proportional increase in age that a two-year-old experiences between birthdays, they would have to wait until they turned 30. Given this view point it’s not surprising that time appears to accelerate as we grow older.

We commonly think of our lives in terms of decades – our 20s, our 30s and so on – which suggests an equal weight to each period. However, on the logarithmic scale, we perceive different periods of time as the same length. The following differences in age would be perceived the same under this theory: five to ten, ten to 20, 20 to 40 and 40 to 80.

I don’t wish to end on a depressing note, but the five-year period you experienced between the ages of five and ten could feel just as long as the period between the ages of 40 and 80.

So get busy. Time flies, whether you’re having fun or not. And it’s flying faster and faster every day.

• Christian Yates is a Lecturer in Mathematical Biology, University of Bath.

.