Tag Archives: Euclidean geometry

The Hounds of Tindalos, Part 3: More Evidence for an Euclidean Form of Life

FBL_CoC                                                     The Hounds of Tindalos on the cover of “Crypt of Cthulhu,” edited by Dr. Robert M. Price. Artwork by <meta http-equiv=”refresh” content=”0; URL=/?_fb_noscript=1″ /> Stephen E. Fabian and shows the “Hounds” as angular or Euclidean forms of life.

Based on what we know about the history of Earth, after the Elder Thing created eukaryotic cells and discarded a large portion of this biological material into the environment of early Earth, life proliferated and speciation occurred through natural selection. Over the history of life on Earth, we know of five mass extinctions with the most well-known being the Cretaceous – Tertiary mass extinction, which occurred approximately 65 million years ago. This is the one that wiped out the dinosaurs. As noted in a previous article this particular mass extinction may have been due to a close encounter with Ghroth, the Harbinger.

The earliest mass extinction was the Ordovician – Silurian mass extinction, which occurred approximately 443 million years ago. During this event about 85% of all sea life went extinct (www.bbc.co.uk), including the trilobites, which were the dominant form of life on Earth at the time. However, not all ancient life was wiped out through mass extinctions. Prior to the first mass extinction, between 635 and 541 million years ago during the Ediacaran Period, the dominant forms of life were an unusual group of sessile (stationary) organisms called rangeomorphs.

Rangeomorphs_www.livescience.com       Rangeomorphs were an ancient group of fractal organisms that dominated the primeval seas approximately 600 million years ago (www.livescience.com).

During the Ediacaran Period, the dominant form of life on Earth – the rangeomorphs – had an extreme form of Euclidean, morphological geometry that exhibit self-similarity or repeated patterns at various scales; such objects are said to have fractal dimension. While straight lines and classical Euclidean geometry is relatively rare in nature, fractal geometry is still present. An example of this is the shape and morphology of fern (see below). Other examples include the shapes of snowflakes or naturally forming crystals. Back in the Ediacaran Period fractal organisms dominated the nutrient-rich seas. Unfortunately for them, the eventual development of mobile organisms that can hunt or graze upon benthic creatures that remain stationary is hypothesized to be responsible for their extinction (www.livescience.com). Thus, essentially these fractal animals that lived as plants in the end went extinct once they became prey for more mobile organisms.

HofT_KrisztianHartmann_ArtStation.com                               Hound of Tindalos by Krisztian Hartmann (www.ArtStation.com)

While the Hounds of Tindalos are not the product of Terran evolution, nor are they even residents of our Universe, they may represent an extreme fractal form of life from another Universe. As I previously suggested curves and non-Euclidean geometry tend to be more common in nature than classical Euclidean geometry. However, the Hounds may originate from a fractal Universe where curves are as alien to them as the third dimension is to a two dimensional “flatlander.”

Carl_Sagan                Carl Sagan explaining how it would be difficult for a 2 dimensional being to perceive the true nature of a 3 dimensional being. The same may be said about the Hounds of Tindalos – can they even perceive curves and non-Euclidean geometry? Is that is the case, can we really perceive their true nature?

I would like to conclude this article by letting those know who have submitted questions, statements or hypotheses that I will be responding to them shortly.  I’ve been very busy this summer but will respond.  Many of you have stated that straight lines are more common in nature than I conveyed in the previous article.  Yes, I would agree that straight lines are present in nature, and in fact Euclidean geometry is present as in the case of fractals such as snowflakes and ferns; however, curvatures and non-Euclidean geometry is still more common and pervasive that Euclidean geometry (at least in nature). Indeed, a lot of what we perceive as a straight line is in fact not. For example, while the horizon of where the sea means the sky may be perceived as a straight line it is in fact curved. The curvature of the Earth is why ships seem to “sink” into the sea as opposed to keep getting smaller and smaller until the disappear.  Another example would be sunlight – some would say sunlight is a straight line when in reality is actually a dual form of energy as both a photon particle and a wave, which is not Euclidean.  I need to confirm this but I don’t think much of quantum mechanics could be represented or explained as Euclidean geometry. In any event, I must admit that I probably over-hyped the point that Euclidean geometry is rare in nature. While it may not be rare in nature, it is certainly not as common as non-Euclidean geometry.

HofT_kingovrats.dev                               Hound of Tindalos by KingOvRats (www.deviantart.com)

Next time we will discuss the truly alien biology of the Hounds themselves.  Thank you – Fred.

The Hounds of Tindalos, Part 2: Residents from an Euclidean Space-Time

tindalos3_TorenAtkinson_gwthomas.org                     Hounds of Tindalos by Toren Atkinson (www.gwthomas.org)

“Time is merely our imperfect perception of a new dimension of space. Time and motion are both illusions. Everything that has existed from the beginning of the world exists now. Events that occurred centuries ago on this planet continue to exist in another dimension of space. Events that will occur centuries from now exist already. We cannot perceive their existence because we cannot enter the dimension of space that contains them.”

This passage from Frank Belknap Long’s “The Hounds of Tindalos” may be a direct reference to the existence of multiverses, stitched or linked together not spatially but on a temporal basis, through time.  Some evidence suggests that space-time was created as a result of the Big Bang.  Additionally, while these universes may be completely inaccessible to one another, at least with what science currently tells us, they may have the common thread of time (see the lower right hand section of illustration below).  Thus, if the Hounds can travel outside of space and time, they may have the ability to move from one multiverse to another.

big_bang_nageo From National Geographic

In the same paragraph Long does go on to say, “Human beings as we know them are merely fractions, infinitesimally small fractions of one enormous whole. Every human being is linked with all the life that has preceded him on this planet. All of his ancestors are parts of him. Only time separates him from his forebears, and time is an illusion that does not exist.”

This second half of the paragraph sounds like some of the text that Clark Ashton Smith used to document the dissolution of Paul Tregardis in “Ubbo-Sathla.” As Paul gazed into a strange orb-like crystal he, “…seemed to live unnumbered lives, to die myriad deaths, forgetting each time the death and life that had gone before.” Also, “Life by life, era by era, he retraced the long and groping cycles…” These passages indicate that they may be a way for humans to remove themselves out of our space-time and/or perceive the multiverses at the same time, in effect traveling through time. In the case of Paul Tregardis this was accomplished through the use of a strange, milky orb-like crystal, while in the “Hounds of Tindalos” Halpin Chalmers accomplished this with the use of the drug Liao. In each case there was a terrible price to pay. However, these similarities of traveling through space-time through the interconnectedness of a specific entity does beg the question – is there some relationship between Ubbo-Sathla and the Hounds of Tindalos?


Is there a connection between the Hounds and Ubbo-Sathla? (www.brawlofcthulhu.blogspot.com)

One of the main points made about the Hounds is that they are residents of “angular time.” Humans, our world and our universe is all structured as curved space-time. While this is an extreme oversimplification, our three-dimensional space is curved in the fourth dimension of time (Nothing: A Very Short Introduction by Frank Close; 2009). As Einstein demonstrated the gravity associated with mass in universe bends or curves light and space. Gravity is also the reason way planets and stars are spherical. In addition, nature itself is very curved.  For example, cells are spherical and details measurements of the human brain currently being conducting are using tools associated with curvilinear space.  Straight lines are somewhat rare nature while curves clearly dominate the nature world (see below). Another way of describing our curvilinear universe is by calling it non-Euclidean. Thus, we are very non-Euclidean, curvilinear organisms.



curves Examples of the non-Euclidean structure of nature

While nature is very non-Euclidean humans are far more comfortable in Euclidean geometry and engineering. The Euclidean approach to design and building has served our civilization well over the centuries, through the building of the pyramids to the construction of bridges to the design of space probes, we depend on Euclidean mathematics and geometry in our everyday life. We depend on it so much so that when confronted with the non-Euclidean geometry of Cthulhu’s city of R’lyeh we are disoriented and confused (of course complicating this are other distortions in our space-time relative to Cthulhu itself but that is another discussion).

r_lyeh_by_moonlight_by_2manyworlds-d910yx7.png R’lyeh by Moonlight (by 2manyworlds found at http://www.deviantart.com)

So what does all of this have to do with the Hounds of Tindalos?  Based on Chalmers use of the drug Liao and his investigations, the Hounds are a form of Euclidean life that is very different from organisms within our curvilinear universe. Thus, while we exist in a curved space-time, the Hounds must reside within an angular (Euclidean) space-time. Chalmers called our curved universe “pure” while the angular universe as “foul.” I think this was more of an anthropomorphic attempt to express the strangeness of the Hounds’ angular, Euclidean universe.

Hannes_Bok_1946_ArkhamHouse Hannes Bok’s interpretation of the Hounds of Tindalos may have convey their true angular, Euclidean structure and biology

If the Hounds are essentially a form of angular life, this would contribute toward the idea that they live outside of our space-time and can only enter our reality through angles. In a strange way, our Euclidean approach to architecture and engineering has made it easier for the Hounds to enter our reality. As our civilization has become more technologically advanced and continues to move more away from nature, the more Euclidean we have become. Think of all of the buildings, structures, machines and devices that now exist. From the world’s largest buildings to the smallest personal devices, there are a countless number of angles for the Hounds of Tindalos to enter our space-time. Is this why many of the Old Ones (e.g. Cthulhu) and other alien civilizations (e.g. Elder Things, Mi-Go) avoid the use of Euclidean geometry? To avoid opening a portal between curved space-times and angular space-times? If the portal was large enough would a collision between universes create another Big Bang? Chilling questions to say the least.

Next time we will talk about the “biology” of the angular, Euclidean Hounds of Tindalos. Thank you – Fred.

A Tale of Two Lovecraftian Cities


R’lyeh by the artist Mr. Loach.

H.P. Lovecraft used the term non-Euclidean in a few of his stories including  “The Call of Cthulhu” and “Dreams in the Witch House.”  In specific reference to “The Call of Cthulhu” the term non-Euclidean geometry is used to describe Cthulhu’s sunken City of R’lyeh.  However, the term non-Euclidean was not used to describe the great cities of the Elder Ones in “At the Mountains of Madness.”  This article compares these two alien cities to one another and discusses the non-Euclidean nature of R’lyeh.

I have already discussed what Euclidean and non-Euclidean means in a pervious article but for the sake of this discussion these terms will be briefly reviewed.  Simply put the term Euclidean refers to 2-dimenional (squares, triangles and circles on a plane) and 3-dimenional (cubes, pyramids and spheres in space) realities.  Human architecture is almost entirely based on Euclidean geometry (see below).


Euclidean three-dimensional space (from http://www.wikipedia.org)

While human architecture may be heavily Euclidean, other components of our lives are dependent on non-Euclidean geometry, such as the use of Global Positioning System (GPS) technology due to the curvature of the Earth (see below).  In addition, much of nature is non-Euclidean in design.


A comparison between Euclidean and non-Euclidean (elliptic and hyperbolic) Geometries (www.blendspace.com)

From a Lovecraftian perspective this may seem a little disappointing, however, shown below is an example of non-Euclidean architecture.  Such designs can be a little disorienting but as will be discussed in more detail below, based on HPL’s text I hypothesize that the non-Euclidean description of R’lyeh is only a partial attempt to understand the truly alien aspect of the city.  However, before we discuss R’lyeh in more detail, I want to briefly review the Elder Ones cities in “At the Mountains of Madness.”


 A truly non-Euclidean view of R’lyeh (www.jennytso.com)

HouseinAbiko_RenovationsofNationalExhibitionCentre6 House in Abiko, from Renovations of the National Exhibition Centre – an example another variety of non-Euclidean architecture.

The cites of the Elder Things in “At the Mountains of Madness” were truly strange and alien, being described as “…curious regularities of the higher mountain skyline – regularities like clinging fragments of perfect cubes…” and “…no architecture known to man or to human imagination, with vast aggregations of night-black masonry embodying monstrous perversions of geometrical laws and attaining the most grotesque extremes of sinister bizarrerie.”  Other terms used to describe the alien Elder Ones cities included truncated cones, tall cylindrical shafts bulbously enlarged and often capped with tiers of thinnish scalloped discs. multitudinous rectangular slabs or circular plates of five-pointed stars, cones and pyramids either alone or on top of other cubes or cylinders some of which were flatted on the top, and needle-like spires in clusters of five.

At the Mountains of Madness by Stephan Mcleroy (www.stephenmcleroy.com)

While the descriptions of the Elder Ones cites are indeed alien, they are primarily Euclidean in nature (e.g. cubes, cylinders, etc.) but with some small inclusion of non-Euclidean architecture.  More importantly, they did not give the impression of a geometry being “all wrong” as Wilcox described R’lyeh in his dreams or the dream-place geometry, extra-dimensional impression Johansen had when he landed on the island.

R’lyeh by the great artist John Coulthart (www.johncoulthart.com)

R’lyeh appears to be more “alien” to us relationship to the cities of the Elder Ones, which corresponds with our biological relationship between the Elder Ones and Cthulhu (including its spawn).  Essentially, HPL was very explicit in stating that the Elder Ones, while being very alien, were still made of the same matter we are; we are residents of the same universe.  In contrast, Cthulhu and its spawn are well known to be extra-dimensional entities.  They are not of this universe and are not composed of the same matter we are.  Thus, their manifestations into our reality is more than likely not their “true” form – simply an interpretation of their appearance in a three dimensional / one time universe.  Sort of the way you can draw a representation of a cubic on a sheet of paper.  It is an interpretation of a three dimensional object on a two dimensional plane.


This is a drawing of a cube, interpreting what a 3-dimenional object looks like on a 2-dimensional plane.

Since Cthulhu and its spawn are extra-dimensional, their architecture is more than likely extra-dimensional as well.  This would explain why the geometry of the R’lyeh just does not feel “right” to humans.  Being creatures of 3 dimensions and 1-time scale, our senses and previous experiences are making an attempt to perceive Cthulhu and R’lyeh.  Sometimes our senses clearly get this extra-dimensionality wrong such as when Parker was swallowed up by an angle of masonry that was acute but behaved as if it were obtuse as documented in the end of “The Call of Cthulhu.”


R’lyeh by Pal Carrick

To conclude, while the cities in Antarctica are clearly alien, they were built by the Elder Ones, creatures of our universe and reality.  In contrast, R’lyeh seems more alien and “wrong” since it is only a representative manifestation of what it looks like in our reality.  Thus, our perception of what is looks like is very different than what is actually looks like in its own multi-dimensional reality.  In fact, since we are limited to 3 dimensions and 1 time we can never know what this multi-dimensional city truly looks like.  This goes for its extra-dimensional denizens, which includes Cthulhu and its spawn.  However, if we could somehow alter, expand and/or increase our senses, maybe we could then see the true form of both Cthulhu and R’lyeh.

Next time we will expand on the concepts of extra-dimensionality, with specific discussions on Cthulhu itself.  Thank you – Fred.


R’lyeh by Decepticoin (www.deviantart.com)

Lovecraft and Mathematics: Non-Euclidean Geometry

Over the next few articles I will be discussing how HPL incorporated mathematics and physics into his fiction.  However, other subjects, such as astronomy and biology, may crop up from time to time.  Sometime in February discussions will begin on the Yithians.  For this article the focus will be on “non-Euclidean geometry”.

                        Brown University’s Ladd Observatory in Providence, RI

While HPL loved astronomy, he was not a fan of mathematics.  As a student, he thought of having a professional career in chemistry or astronomy but the difficulties he had with algebra made him realize this was not possibility.  In fact, as cited in S.T. Joshi’s I Am Providence: The Life and Times of H.P. Lovecraft (Hippocampus Press, 2013), HPL remarked in 1931,

“In studies I was not bad – except for mathematics, which repelled and exhausted me.  I passed in these subjects – but just about that.  Or rather it was algebra which formed the bugbear.  Geometry was not so bad.  But the whole thing disappointed me bitterly, for I was then intending to pursue astronomy as a career, and of course advance astronomy is simply a mass of mathematics.”

It is interesting to note that HPL was not stratified with the grade he received in Intermediate Algebra during the 1906-07 school year and that he voluntarily re-took the subject the following year.  Unfortunately, based on school records HPL did not receive his high school diploma , finishing only the eleventh grade (Joshi, 2013).  Never completing high school and going to college was always a personal failure in his mind, which he mentions a number of times.  However, HPL clearly had a strong interest in the sciences throughout his life and this was well engrained in his stories, incorporating the most up-to-date scientific knowledge at the time into his writings.

Lovecraft by Greg Nemec

Upcoming articles will go into more detail in the use of mathematics in HPL’s work, however, for this article I wanted to focus on one of his more popular phrases:

Non-Euclidean Geometry– to understand what non-Euclidean geometry is you have to know what is meant by Euclidean geometry.  Euclid was a Greek, born around 300 B.C. and his best known for developing the  math of geometry.  Part of his treatise The Elements, included a series of axioms and notions which laid the foundation for modern geometry.  Non-Euclidean Geometry is simply a modification of one of these axioms or notions.  Two of the more “common” types of non-Euclidean Geometry are hyperbolic geometry and elliptic geometry.  For convenience, each type of geometry is shown below:

      From http://www.mathforum.org

Thus, Euclidean space is essentially “planar” geometry.  Straight lines, squares, cubes and angles of 45 and 90 degrees – this is Euclidean geometry and to someone like HPL who was so interested in architecture, nothing exemplified human civilization than their buildings and infrastructure.  Non-Euclidean geometry is not just triangles and squares – the familiar circle formulas of C = 2 π r and A = π r(C = circumference; r = radius; A = area; π is 3.14159…) are very Euclidean.  Thus, the elliptical and hyperbolic geometries do not follow these Euclidean circle formulas and thus are defined as non-Euclidean geometry.

When I first figured this out in high school or college I was sort of disappointed because in my mind Lovecraft’s non-Euclidean geometry looked far more alien than elliptical and hyperbolic geometries however, stop for a moment and think about seeing an entire city built using this type of geometry.  Shown below are some varying examples:

euclidean-geometry-vs-non-euclidean-geometry_www.jennytso.comFrom http://www.jennytso.com

Cover art for The Art of H.P. Lovecraft’s Cthulhu Mythos (by Michael Komarck)

                                                       R’lyeh by John Coulthart (www.johncoulthart.com)

I love these examples of Lovecraft’s non-Euclidean geometry, particularly in showing what R’lyeh looks like.  All three exude a very alien impression.  However, in my mind R’lyeh would look far more alien.  For example, I see the buildings and structures actually changing shape simply by viewing them from different points of view – something that is obviously very difficult to show in a drawing or painting.  What is truly amazing is how in simply using an unusual phase such a “non-Euclidean geometry” Lovecraft was able to stimulate the imagination of his readers as well as other writers and artists generations later.

Next time I will be discussing the use of mathematics in Lovecraft’s The Dreams in the Witch House.  Thank you – Fred