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Yemelyan Pavlov
Yemelyan Pavlov

Euclid's Window : The Story Of Geometry From Pa...


Through Euclid's Window Leonard Mlodinow brilliantly and delightfully leads us on a journey through five revolutions in geometry, from the Greek concept of parallel lines to the latest notions of hyperspace. Here is an altogether new, refreshing, alternative history of math revealing how simple questions anyone might ask about space -- in the living room or in some other galaxy -- have been the hidden engine of the highest achievements in science and technology. Based on Mlodinow's extensive historical research; his studies alongside colleagues such as Richard Feynman and Kip Thorne; and interviews with leading physicists and mathematicians such as Murray Gell-Mann, Edward Witten, and Brian Greene, Euclid's Window is an extraordinary blend of rigorous, authoritative investigation and accessible, good-humored storytelling that makes a stunningly original argument asserting the primacy of geometry. For those who have looked through Euclid's Window, no space, no thing, and no time will ever be quite the same.




Euclid's Window : The Story of Geometry from Pa...



Finally, while The Elements may or may not be, as Dodgson believed, the perfect text for teaching plane geometry, it is unique in the history of our species. It may well be the most read, most published book in history, outstripping even the various religious documents ([7], p. 55). For that reason alone, quite apart from its intrinsic value to mathematics, we should continue to find ways to pass it on to each successive generation. This is our contribution to that cause.


Abraham Lincoln now rests in Oak Ridge Cemetery in Springfield, Illinois. There, he lies, with his wife and three of his sons, as perhaps the most revered figure in American history. He rose from obscurity, worked hard to achieve greatness, guided the ship of state through its most critical period and gave his life for his country. Truly, his life was the story of the United States of America.


Euclid of Alexandria.Euclid, who flourished around 300 BC, is known to mosthigh school students as the father of geometry.Surprisingly little is known of his life, not even his dates orbirthplace. Shortly before 300 BC, Ptolemy I foundedthe great university at Alexandria, the first institution of its kind,and not unlike the universities of today. Euclid was recruited,probably from Athens, to head the mathematics department.


Two famous stories are told about Euclid. It is said that Ptolemyasked him if geometry could be learned without reading the Elements, to which Euclid replied, "There is no royal road togeometry.'' (This story is also told aboutMenaechmus and Alexander the Great, which perhaps diminishes its credibility somewhat.) Inresponse to a student who questioned the use of geometry, Euclidreportedly ordered that the student be given three pence, "since hemust needs make gain of what he learns.''


Geometry is a branch of mathematics that deals with lines, shapes, points, curves, and surfaces. The word geometry is derived from two Greek words: geo, meaning earth, and metrein, meaning to measure. Not one singular person discovered geometry. Multiple people discovered aspects and concepts that helped evolve and grow our knowledge of geometry over time, starting with ancient ages to current, modern times.


Geometry surrounds us every day. We are surrounded by shapes, lines, points, and dimensions every day. At school, work, and in our houses. The men and women who are building the next skyscraper use geometry to decide how tall to build the building, what shapes the windows should be, or how long the interior hallways should be. NASA and those who work in astro science also use geometry. They study planets and stars all day, which are shapes. They are trying to discover the area and surface of outer space.


Many years before the Greeks discovered these geometric ideas and gave them names, Indian culture used geometry as a part of their culture. In Indian culture, they worship many Hindu gods and create and construct altars to honor them. While constructing these altars, they were actually using geometry. The Sulba Sutras, written from around 800 BC to 200 BC, are instructions on how to construct these altars. The people of the Hindu faith want to make their Gods happy and proud. To do so, they followed the instructions of the Sulba Sutras very carefully, so mathematical veracity was of high importance. The Sulba Sutras do not provide any proof, unlike Euclid and The Elements. In the Sulba Sutras, instructions on how to construct circles, squares, rectangles, and triangles for their altars were the unknown beginnings of the use of geometry. They did however include ideas such as constructing a square of area equal to a given rectangle. However, the biggest connection to geometry included in the Sulba Sutras was its instructions on ropes measuring the altars: The rope which is stretched along the length of the diagonal of a rectangle produces an area which the vertical and horizontal sides make together. Simply put, this is almost identical to the Pythagorean Theorem.


Non-Euclidean geometry is different from Euclidean because it consists of curves instead of straight lines. The basis of Non-Euclidean geometry stems from Euclid's fifth postulate in The Elements: That, if a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles. The postulate is only true in regular two-dimensional Euclidean geometry, not in three dimensions. Because geometry has been studied for thousands of years, it has been deduced that Euclid's fifth postulate is not always true and therefore formed another small branch of geometry dealing with three dimensions and curved lines.


No one man founded geometry. It is a section of mathematics that has evolved with the help of many different people from all over the globe. Many Greeks gave their names to certain theorems and ideas, but it should be known that many of those ideas were discovered by Chinese and Indian cultures as they built tombs and altars for Gods and loved ones. For thousands of years, more aspects of geometry have been identified and proven. Because of those discoveries, the world has evolved. Because of lines and distance, maps and GPS systems have been created. Because of shapes and dimensions, X-Ray machines and video games have been created. Geometry is the basis of everything we do.


Geometry surrounds us every day. It helps us tell direction and gets us from one destination to the next. It assists us in understanding the space around us. No matter what job or profession a person has, the need to understand space, area and numbers will most likely be involved. That's geometry.


Nature themes can be found in the earliest human structures: Stylized animals characteristic of the Neolithic Göbekli Tepe; the Egyptian sphinx, or the acanthus leaves adorning Greek temples and their Vitruvian origin story; from the primitive hut to the delicate, leafy filigrees of Rococo design. Representations of animals and plants have long been used for decorative and symbolic ornamentation. Beyond representation, cultures around the world have long brought nature into homes and public spaces. Classic examples include the garden courtyards of the Alhambra in Spain, porcelain fish bowls in ancient China, the aviary in Teotihuacan (ancient Mexico City), bonsai in Japanese homes, papyrus ponds in the homes of Egyptian nobles, the cottage garden in medieval Germany, or the elusive hanging gardens of Babylon.


In rural environments, human-nature interactions are abundant, and this regular exposure to nature has restorative qualities that we perhaps take for granted. Suburban settings are typically rife with intuitively applied biophilic design; the suburban yard with shade trees, grass, low shrubs, and beds of flowers is essentially an analogue of the African savanna. Porches and balconies offer more than just quaintness and real estate value; many suburban homes and urban rowhouses are raised 18 inches or more, creating a Prospect-Refuge condition with views from windows, stoops and porches. The potential human health benefits are undervalued in high-density settings where residential towers with balconies are both limited and only available to high-paying tenants.


The Dockside Green community on Vancouver Island, Victoria, BC Canada, is a great example of non-rhythmic stimuli. The implementation of habitat restoration and rainwater management has led to ephemeral experiences of swaying grasses, falling water and the buzz of passing insects and animals that are visible from walkways, porches, and windows around the community.


  • Live-Action TV Doctor Who: The Expanded Universe speculates that this is the default setting for the interior of a TARDIS, and that the Doctor's TARDIS projects a more easily comprehended interior so as not to freak out the Doctor's human companions. She is just a sweet old thing. Even if the above speculation is incorrect, the Bigger on the Inside dimensions of the TARDIS are occasionally enough to disturb someone, most memorably with Jackson Lake's mild panic-attack in "The Next Doctor". From his reaction, it was giving him claustrophobia and agoraphobia at the same time.

  • "Castrovalva": The city of Castrovalva itself is built like this, as part of a trap to destroy the Doctor. It appears perfectly normal, but if you try to leave the centre of town, no matter what direction you travel, you'll soon end up back there. When the city starts breaking down, it begins to resemble an Escher picture.

  • "The Lodger": The Doctor uncovers an alien time-distortion device similar to the TARDIS in the upstairs flat of a British apartment building. Amy, poring over the building plans for the address, discovered that the building didn't even have an upstairs, it was a one-story building. Perception filters kept people from noticing anything out of the ordinary.

  • "Journey to the Centre of the TARDIS": while the TARDIS interior looks like a fairly ordinary albeit vast spaceship, the TARDIS is revealed to be capable of tying its internal spatial and temporal dimensions into knots. At one point it threatens to do this to trap some thieves inside forever to stop them stealing some of its technology. This is something the TV show occasionally alluded to in the past with the TARDIS being able to delete and move rooms about and having an "unstable pedestrian infrastructure", and novels, comics, and audios have expanded on this for years, but this episode marks the first time we've actually seen it first hand.

  • In the Night Garden... is a BBC kids' show (from the people who made Teletubbies) where the various characters often ride around the eponymous garden in the Ninky-Nonk (a train without tracks) or the Pinky-Ponk (an airship). When they're boarding, these vehicles are comfortably large enough to accommodate all of them, yet when they're actually in motion the Ninky-Nonk is small enough to run up trees and over branches, and the Pinky-Ponk is small enough to get knocked off course by a toy ball.

  • Neverwhere does a very nice demonstration of this in passing. The protagonist is led down into the London Underground, then through a door, and down a stair case. This continues, always going down, until they reach a small door and step out on to the roof of a building.

  • In Rose Red, the titular mansion is like this. Sometimes. It was built to perfectly normal standards, but after a series of incidents it went from "just" haunted to something more, and may in fact have been sentient. Features include staircases leading into ceilings, dead-end hallways that screw with perspective, rooms that weren't there a minute ago (or were there but aren't any more), and other hilarities. About the only guaranteed stable locations are the entryway, the attic and the arboretum, and even then the things in them often are moved around or fully animate.

  • In Severance (2022), the offices of Lumon Corporation are just...off. The building already looks unsettling on the outside, with its weirdly angled mirrored parking lots. On the inside it has endless white hallways, dark elevators that only go down, giant (but low-ceilinged) offices with only a few desks in them, and an entire full-scale replica of Kier Eagan's house.

  • Stargate SG-1: The spacecraft used by the Goa'uld are relatively normal... until you notice the pyramid on top. Naturally, the entire spaceships can fold up so that their central pyramid can land on a planet-bound pyramid. Not to mention how a triangular-pyramid-shaped spacecraft can land on a square-pyramid.

  • The plot of the (admirably silly) Star Trek: Voyager episode "Twisted" where the ship becomes a maze where no door or hallway leads the same place twice due to a Negative Space Wedgie.

  • Similarly, in Star Trek: TNG "Where Silence Has Lease", the Enterprise winds up in a Negative Space Wedgie and Genius Loci where physics goes right out the window. They drop a beacon and head directly away from it only to find themselves heading directly toward it, they explore another ship where leaving a room results in re-entering the room from a different door, etc.

  • Threshold involved an alien invasion. The aliens used devices that apparently contained more that four dimensions, and cannot be fully perceived visually. Just seeing or hearing the signals originating from these "beacons" can kill or transform the view into an alien agent, with triple DNA helix where earthlife has only contains double. The aliens themselves are usually seen in dreams; crystal forests where spider-like entities are only partially seen.

  • In The Twilight Zone episode "Little Girl Lost", a little girl falls through a portal in the wall of her bedroom into an alternate dimension, in which space is twisted, distorted and nonsensical to ordinary human perception. Fortunately, the family dog's superior hearing and sense of smell help get the little girl back into our dimension before the portal closes forever.

  • In Warehouse 13, the personal effects of permanent prisoners of the Warehouse are stored in the Escher Vault, which is basically a three-dimensional M.C. Escher painting. Authorized personnel use special goggles to follow along with the vault's ever-shifting perspective. Unauthorized personnel are never seen again unless they have Super Speed.

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