Illusions

Geometrical illusions are examples of how our mind tries to find orderly representation out of ambiguous, disordered 2D images. The images transmitted to our brain from our retina are an imperfect representation of reality; our visual system is capable of processing the information received from our eyes to extract meaningful perceptions. However, this can sometimes go wrong, leading to faulty perceptions.

Illusion of Position: Poggendorff Illusion

The Poggendorff illusion is an image where thin diagonal lines are positioned at an angle behind wider stripes. The blue line on the right appears to line up with the black line on the left, however in actuality, the black and red lines match up.

File:Poggendorff illusion.svg
Source: newworldencyclopedia.org

Illusion of Length: Müller-Lyer illusion

The Muller-Lyer illusion is a well-known optical illusion in which two lines of the same length appear to be of different lengths.

Source: Wikipedia

Illusions of Orientation

Zöllner illusion

In this illusion, the black lines do not seem to be parallel, but in reality they are.

 

Source: Wikipedia

 

Café Wall illusion

This is an optical illusion in which parallel straight dividing lines, between staggered rows with alternating black and white ‘bricks’, appear to be sloped.

Source: Wikipedia

Illusion of Size: Delboeuf Illusion

Two circles of identical size are placed near to each other and one is surrounded by an annulus (ring-shaped object). The surrounded circle appears larger than the non-surrounded circle if the annulus is close, while appearing smaller than the non-surrounded circle if the annulus is distant.

Source: foodnavigator-usa.com

Illusion of the Straightness of Lines: Hering Illusion

When two straight and parallel lines are placed in front of radial background, the lines appear as if they were bowed outwards.

Source: Wikipedia

Illusions of Vertical/Horizontal Anisotropy

The vertical–horizontal illusion demonstrates the tendency for observers to overestimate the length of a vertical line relative to a horizontal line of the same length.

Source: Wikipedia

Impossible Objects

Impossible objects consist of 2D figures which are subconsciously interpreted by our visual system as representing a projection of a 3D object.

Penrose Triangle

Although it was first created by the Swedish artist Oscar Reutersvärd in 1934, Lionel and Roger Penrose independently devised and popularised the Penrose triangle in the 1950s, describing it as “impossibility in its purest form”.

1124-i
Source: cutoutfoldup.com

Penrose Staircase

The Penrose staircase is a variation on the Penrose triangle, and is a two-dimensional depiction of a staircase that seems to form a continuous loop, so that a person could climb them forever and never get any higher. This is clearly impossible in three dimensions.

Source: Wikipedia

Impossible Fork/Blivet

The impossible fork appears to have three cylindrical prongs at one end which then mysteriously transform into two rectangular prongs at the other end.

File:Poiuyt--opaque.svg
Source: Wikipedia

Sources: 1 | 2

Hope you enjoyed this round up of some mathematical optical illusions! M x

 

Densest Packing Problem Breakthrough

As suggested by Gaurish4Math, in todays post I’ll be discussing the recent breakthrough in the sphere packing problem.

What is the Sphere Packing Problem?

A sphere packing is an arrangement of spheres that do not overlap within a containing space. The problem is to find the best packing of spheres in dimension n. The densest packing of spheres was only known in dimensions 0, 1, 2 and 3, before this breakthrough.

E8 Lattice

The E8 lattice can be characterised as the “unique, positive-definite, even, uni-modular lattice of rank 8”. It is formed by taking all the sums of the vectors in the root system. This root system contains all roots (a1,a2,a3,a4,a5,a6,a7,a8) where all ai are integers or all ai are integers plus 1/2, the sum is an even integer, and sum of the squares is 2.

It can be expressed as follows:

36dc5a84807b5311e8372148687191a1.png

Breakthrough

Maryna Viazovska published a paper proving that if you centre spheres at the points of the E8 lattice, you get the densest packing of spheres in 8 dimensions. Following this, Viazovska joined Cohn, Kumar, Miller and Radchenko to prove that the Leech lattice gives the densest packing of spheres in 24 dimensions.

The starting point for Viazovska’s breakthrough was a method developed by Cohn and Elkies in 2001 that improved the upper bounds to the density in dimensions 4-31, giving extremely good results for dimensions 8 and 24; they showed that the densest packing in 8 dimensions could be no more than 1.000001 times as dense as that coming from the E8 lattice (Viazovska then reduced this to 1). They showed that, provided a function satisfies a number of conditions, it will give the upper bound on the density. The problem was to find this function.

However, finding the right function proves enigmatic and to find it Maryna had find Fourier transforms of some modular forms and prove certain estimates about them.

Congratulations to Viazovska, Cohn, Kumar, Miller and Radchenko for this fantastic achievement!

Sources: 1 | 2 | 3

Apologies that this post is quite short – the mathematics involved is quite complex and hard to resume in a post. Hope you enjoyed it! Mx

Hyperbolic Geometry

The man made world we see around us is constructed using straight lines, from the houses we live in to the skyscrapers we admire. However, in nature we observe wonderful shapes such as the beautiful undulations of coral and the crinkled surface of lettuce.

These natural phenomena follow hyperbolic geometry, where the plane is not necessarily flat, as opposed to the conventional Euclidean geometry.

Hyperbolic Geometry

Hyperbolic geometry is not considered Euclidean as it violates one of the axioms called the parallel postulate:

“If a line segment intersects two straight lines forming two interior angles on the same side that sum to less than two right angles, then the two lines, if extended indefinitely, meet on that side on which the angles sum to less than two right angles.”

Put more simply, this axiom says that there is only one possible straight line through a point that does not meet the original line, as depicted by the image below.

Corals, crochet and the cosmos: how hyperbolic geometry pervades the universe
Source: phys.org

 

However, this is untrue in hyperbolic geometry – there can be more than one line passing through this point that do not meet.

Corals, crochet and the cosmos: how hyperbolic geometry pervades the universe
Source: phys.org

When this non-Euclidean geometry was discovered in the early 19th century, it was not easily accepted by mathematicians. Wolfgang Bolyai, a Hungarian mathematician, said to his son János Bolyai: “for God’s sake, please give it up!”

Hyperbolic Geometry in our Universe

In nature, we can see a variety of different manifestations of hyperbolic geometry, including corals and reef organisms like kelp. Scientists believe that this is due to the fact in organisms that have to maximise their surface area (such as in filter feeding animals), hyperbolic shapes provide an excellent solution.

In 1997, physicist Andrei Rode also built hyperbolic surfaces at a molecular scale from carbon atoms to create carbon nano-foams.

Furthermore, although the shape of our universe is still unknown, it is a topic of great research with instruments such as the Hubble Space telescope. Some evidence suggests that our universe is an Euclidean structure, however new evidence indicates that we might just live in a hyperbolic world.

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I’m sorry that today’s post is quite a short one. I will resume my ‘Forgotten Mathematicians’ series on Friday! Also, I’m trying to add more of my sources in my posts, so let me know what you think! M x

Pictures of Maths III

This is the third installment of my ‘Pictures of Maths’ series. Hope you enjoy!

Symmetry in Honeycomb

The structure of honeycomb displays symmetry.

The complex folding patterns that arise when layered paper is put into a test machine and squashed.

The complex folding patterns that arise when layered paper is put into a test machine and squashed.

A sample image generated by Lawrence Ball's harmonic maths.

A sample image generated by Lawrence Ball’s harmonic maths.

 

geometric art in ceiling of building

Geometric art is often used as decoration in the ceilings of buildings.

Vitruvian_Man

‘Vitruvian Man’, drawn in 1487 by Leonardo Da Vinci, showed the relationship between the human body and geometry. It is a piece of art that represents how closely connected science and art are.

look at blog post for more info

The 421 polytope is believed to be the most geometrically complex and aesthetically beautiful structure in mathematics. It is the algebraic form at the centre of a Universal Theory of Everything. It was originally describe in the late 19th century and models all interactions and transformations between known and theorised sub-atomic particles. The theory is an attempt to unify quantum physics and gravitation in hopes of ultimately explaining the fabric of the universe. The visualisation was hand drawn in illustrator to an accuracy of 1/10,000 of a millimetre.

Klein bottle

The Klein Bottle: an object with no boundaries, no inside or outside. It is a one sided, non-orientable surface. That’s topology for you!

Fractal Structure in Ferns

Fractal patterns are visible throughout nature, for example in ferns as displayed above.

Let me know which one is your favourite! Mx

 

Art and Maths: Connected Throughout History

For thousands of years, artists have used mathematical concepts in their work. In this post, I will quickly reveal some connections between these two fields throughout history.

Golden Ratio

The golden ratio is roughly equal to 1.618. The special nature of this ratio appealed to the Greeks, who thought that objects in this proportion were particularly aesthetically pleasing. It has been said that they used this ratio in their architecture and statues to ensure their beauty, for example the dimensions of the Parthenon. In fact, throughout history there have been a number of pieces of art that exhibit the golden ratio: Leonardo Da Vinci’s paintings or Michelangelo’s David. However, it has been debated whether Ancient or Renaissance artists consciously used this ratio, or whether it is simply a numerological coincidence.

Geometric Patterns

Geometric patterns – simple arrangements of mathematical shapes and figures – have been widely used in decoration throughout history. For example, the ‘Flower of Life’ pattern was used on the Temple of Osiris at Abydos in Egypt. Dating back about 5000 years, it consists of circles positioned in rows, each one centred on the circumference of circles in neighbouring rows.

Additionally, Mosques throughout the world are embellished with elaborate geometrical patterns, which symbolize the divine order of the Universe. The use of the geometrical patterns is due to the fact that Islamic art traditionally does not depict people and animals.

mosque

Tessellations

Popularised by Maurits Escher, tessellations are one of the more well-known and direct forms of mathematics in artwork. A tessellation is a tiling of a geometric shape with no overlaps or gaps. Escher made an art form out of colourful patterns of tessellating shapes, including reptiles, birds and fish.

Origami

Origami originated from Japan and is the craft of creating three-dimensional shapes solely by folding paper (usually only one sheet). These shapes range from paper cranes to flowers. If you unfold the piece of paper, there will be a complex geometrical pattern of creases that are made up of triangles and squares. Many of these will be congruent due to the fact that the same fold produced them, revealing the deep links between geometry and ancient art.

Fractals

Fractals are mathematical structures that have the property of ‘self-similarity’, meaning that if you zoom in on one, the same type of structure will keep appearing. I have already talked about extensively in a previous blog post; check it out if you’re interested! (Personally, I find them beautiful).

fractals

Mathematics as Art

The mathematician Jerry King stated, “the keys to mathematics are beauty and elegance and not dullness and technicality”. In ‘A Mathematician’s Apology’ by G.H. Hardy, Hardy explores this idea by explaining his thoughts on the criteria for mathematical beauty: “seriousness, depth, generality, unexpectedness, inevitability, and economy”. Furthermore, Paul Erdos agreed that mathematics had beauty by explaining: “”Why are numbers beautiful? It’s like asking why is Beethoven’s Ninth Symphony beautiful. If you don’t see why, someone can’t tell you. I know numbers are beautiful.”

If this topic interests you, I would highly recommend reading this article in AMS’s Feature Column. M x