Why Does a Soccer Ball Have Patterns of Pentagons and Hexagons on Its Surface
We can define a soccer ball as a spherical polyhedron with the
following properties.
(1) it is a polyhedron that consists only of pentagons and
hexagons;
(2) the sides of each pentagon meet only hexagons; and
(3) the sides of each hexagon alternately meet pentagons
and hexagons.
(4) precisely three edges meet at every vertex
There is
a unique way of putting them together, giving rise to the iconic standard
soccer ball.
The
truncated icosahedron is an Archimedean solid in geometry. It is one of the 13
convex isogonal non-prismatic solids whose 32 faces are two or more types of
regular polygons.[3]
- A soccer ball may appear as a simple sphere, but beneath its smooth exterior lies a meticulously crafted structure.
Do you know that?
Beneath the exterior, there is the artful interplay
of polygons and exquisite craftsmanship.
- Polygons are two-dimensional shapes comprised of straight sides, each forming angles at their vertices.
- The polygons on soccer balls are pentagons and hexagons.
The harmonious arrangement of pentagons and
hexagons transforms a flat pattern into a seamless sphere that mesmerizes
players and spectators.
Every soccer ball has 12 pentagons, and 20 hexagons
come together in perfect harmony.
This symmetrical design ensures the consistent
flight and movement of the ball. With precision stitching and expertise, these
polygons form a tessellation, a tiling pattern that wraps seamlessly around the
spherical surface without gaps or overlaps.
- The sight of a perfectly stitched pattern, whether on the feet of a professional player or in a residential-friendly match, evokes a sense of pride in the rich heritage of sports.
- It is a testament to the timeless beauty of geometry, mastery of design, and craftsmanship.
- It is a captivating blend of art, science, and history, and its elegant symmetries find their place in the heart of the game.
From the delicate arrangement of polygons to the
precision stitching and design, bring the soccer ball to life. We continue to
marvel at the encapsulating magic of mathematics and the elegance of these
spherical wonders.
Pentagons and Hexagons:
- At the heart of every soccer ball, there is a precisely engineered configuration of
pentagons and hexagons.
- The combination of pentagons and hexagons in the ball ensures a balanced and predictable flight during kicks and passes.
Manufacturers have explored alternative arrangements seeking to push the boundaries of traditional patterns.
- The 32-panel design remains embraced and deeply ingrained in the hearts of players and spectators worldwide.
- Every pentagon is surrounded by hexagons.
- These shapes help form the structure of the ball and essentially form a puzzle, when put together, that holds the ball together in a round shape.
Mathematics Behind the Design:
Two-dimensional polygons play a fundamental role in
constructing the intricate pattern. Among them, pentagons and hexagons are the
primary elements of the iconic arrangement. They are arranged in a symmetrical
tessellation, creating a seamless and visually appealing design.
- At the heart of this design lie pentagons and hexagons, working together in perfect harmony to form a spherical structure.
- The relationship between these polygons holds significance in crafting a soccer ball that adheres to the highest standards.
- Ensures that the surface remains smooth and consistent, allowing players to showcase their skills.
So, what is the formula behind the distribution of
pentagons and hexagons on a soccer ball?
The
Goldberg polyhedron GPV (1,1) contains pentagonal and hexagonal faces.
This
geometry is associated with footballs (soccer balls) typically patterned with
white hexagons and black pentagons.
This
structure is the basis of the architecture of Geodesic domes. It also
corresponds to the geometry of the fullerene C60 (buckyball)
molecule.
A
spherical elastic tube inside the outer cover provides the ball with
elasticity. Filling the inner tube with air gives a spherical shape.
Have you
ever noticed?
- The Hexagon panel has alternate long and short sides.
- The hexagon panel is an equiangular hexagon with three longer sides of equal length and three shorter sides of equal length. The confronting sides among the longer and shorter sides are parallel.
The Pentagon has five sides and five vertex angles: n=5
While a hexagon has six sides and six vertex angles: n=6
The sum
of the angle of the polygon= (n - 2)180
For
Pentagon (5 - 2)180 = 540
Each
vertex angle of a Pentagon = 540/5 = 108
The sum
of the angle of the hexagon = (6 - 2)180 = 720
Each
vertex angle of a hexagon =720/6 = 120
To interlock
the Pentagon and hexagon, the sum of the combination of vertex angles of both
must be equal to 360 (sum of the angles at a point = 360)
Sew the
respective borders of 12 pentagon panels and 20 hexagon panels to get a
spherical outer cover.
- One pentagon panel has a connection to five hexagon panels. The Hexagon panel has a connection to three pentagon panels and hexagon panels.
Since
there is no number of combinations that can be equal to 360, there will surely
be a gap. Here comes a suitable explanation, the Geometry of C60.[2]
Geometry
of the fullerene C60:
Construction:
The spatial shape of the C60 molecule is identical to the standard soccer ball polyhedron consisting of 12 pentagons and 20 hexagons.
- It has 60 carbon atoms at the vertices and the edges corresponding to chemical bonds, and precisely three edges meet at every vertex.
- These are the chemical bonding properties of carbon.
- Disjoint pentagons are related to the chemical stability of fullerenes.
- The two infinite families of polyhedra, soccer balls, and fullerenes are in standard soccer balls.
- Thus, properties (1)-(4) together give a unique description of the soccer ball without imposing geometric assumptions.
Application of Euler’s formula:
Euler’s formula, a basic tool in graph theory and
topology, says that in any spherical polyhedron, the number of vertices, v,
minus the number of edges, e, plus the number of faces, f,
equals 2:
v - e + f = 2
Let’s apply Euler’s formula to a polyhedron
consisting of b black pentagons and w white
hexagons.
The total number of faces (f)= b + w
- f=12+20=32.
The pentagons have edges= 5b.
Similarly, the hexagons have a total of edges =.6w
Counted each edge twice because each edge lies in
two different faces.
To compensate for this, divide by 2, and hence the
number of edges is:
- e = (5b + 6w)/2=(12*5+6*20)/2=180/2=90
The number of vertices of pentagon= 5b
The hexagons have vertices = 6w.
In the case of fullerene, assumption (4) says that
each vertex belongs to three different faces. Hence, divide by 3 to compensate:
- v = (5b + 6w)/3 = (60+120)/3=60
Substitute these in Euler’s formula, then we get 2.
The standard soccer ball or buckyball realizes this
minimum value, for which the number v of vertices equals 60,
corresponding to the 60 atoms in the C60 molecule. However, there are many
other mathematical possibilities for fullerene-shaped polyhedra.
What is a buckyball?
12×5=60 new vertices & and 12×5=60 new edges in addition to 30 original edges.
Thus, it has a total of 60+30=90edges.
For a truncated icosahedron, F=12+20=32
E=90 & V=60
which duly satisfies Euler's formula (F+V-E=2). It
is also called Archimedean solid.
The process involves intricate calculations that
strike a delicate balance between geometry and aesthetics.
For determining the number of pentagons, the
formula is relatively straightforward:
- This mathematical precision ensures that each panel fits seamlessly with its neighbors, creating a spherical masterpiece that is equal parts art and science.
- Manufacturers meticulously calculate the placement of each polygon to ensure optimal stability and predictability during gameplay.
Evolution and Technology
Let us explore the historical milestones and the game-changing
impact of technological advancements on soccer ball characteristics and
gameplay.
The
relationship between polygons and soccer ball structure influences the
performance. The configuration of polygons impacts the aerodynamics, speed,
trajectory, and overall behavior of the ball.
This
fascinating geometry-driven innovation is rooted in the history of sports. The
classic 32-panel design, the first soccer ball known as the Telstar, debuted in
the 1970 FIFA World Cup. Since then, it has become an iconic symbol of the
sport.
These are
the measurements for a standard-sized soccer ball used in any FIFA-sanctioned
soccer. However, many different forms of soccer use different variations of the
ball. There are several reasons why soccer balls have both pentagons and
hexagons. However, the main reason has to do with the structure of the ball.
Soccer
balls are round-shaped and should be this way whenever kicked to prevent any
disruption to the match. For this reason, the pentagons and hexagons fit
together like a puzzle when stitched together, ensuring that the ball has
structural integrity.
Another
reason why these panels are important relates to the aerodynamics of the ball.
- With the traditional style of the soccer ball, having both black and white panels, it was easy to separate the players and the field.
- It has 12 regular pentagonal faces, 20 regular hexagonal faces, 60 vertices, and 90 edges.
- The sphere has a circumference between 68 and 70 centimeters. It has a 1.5 percent deviation from sphericity when inflated to a pressure of 0.8 atmospheres.
Does The Number Ever Change?
The measurements involved working out how many
panels are needed. It is hard to understand, but in essence, they relate in a
way they need to fit together in a spherical shape.
So, do you think the maths changes when the size
does?
Yes, when the size of the ball changes, the
measurements involved change too. Having the same number of panels as a
regular-sized ball would not give the ball the same structural integrity.
Who Created The 32-Panel Design?
We know less about the guy who came up with the
idea of a 32-panel design. His name was Eigil Nielsen, a Danish goalkeeper who
also worked for a shoe-making company alongside his soccer career.
Nielsen researched to make a better soccer ball and
worked on the topic for many years before discovering the 32-panel design we
all know and love. He founded Select Sports, a brand that is still going on and
produces some of the best soccer balls in the world.
It took some time for the wider world to become
aware of Nielsen’s creation, but once FIFA got the idea, they wanted to use it
in the World Cup.
- This process creates 12 pentagonal faces and transforms the original 20 triangle faces into regular hexagons. [1][2]
The
pentagons (black) and hexagons (white) capture the iconic image in their
respective colors.
Applications:
The balls used in association football and team
handball are perhaps the best-known example of a spherical polyhedron found in
everyday life. [1]
The ball comprises the same pattern of regular
pentagons and regular hexagons. It is more spherical due to the pressure of the
air inside and the elasticity of the ball.
British traffic signs indicating football grounds
use a uniformly-colored hexagonal tiling section to represent a football rather
than a truncated icosahedron. Mathematician and comedian Matt Parker petitioned
the UK government to change these signs in a geometrically accurate way. They
declined the petition.
Recent developments:
- Techniques for manufacturing balls with sensors that, can track data such as speed, spin, and trajectory.
- Using this data, we can analyze player performance and provide insights for training and coaching.
- Some manufacturers are using recycled plastic or biodegradable materials to reduce the environmental impact of the ball.
- Some researchers are developing materials that can change their properties, such as color, shape, or texture, in response to external stimuli, such as temperature, light, or pressure.
- Some governing bodies are testing new materials that improve visibility, accuracy, and consistency.
- Some manufacturers are using 3D printing or digital printing to create customized or personalized balls.
- Some researchers are developing sensors or chips that can measure and transmit various data, such as speed, spin, and trajectory.
- Some governing bodies are implementing goal-line technology or video assistant referee (VAR) technology to assist the referees in making accurate decisions.
As soccer continues to evolve, so does the science
behind soccer ball design. The mathematics governing polygon arrangements is
ever-green, and researchers and designers seek to push the boundaries of
innovation and optimize performance.
Future developments
Companies such as Umbro, Mitre, Adidas, Nike, Select, and Puma are releasing footballs made out of new materials, expecting more accurate flight and power.[4][5]
References:
1 Kotschick, Dieter (2006). "The Topology and Combinatorics
of Soccer Balls". American Scientist. 94 (4):
350–357. doi:10.1511/2006.60.350.
2 Kotschick,
Dieter (July–August 2006). "The Topology and Combinatorics of Soccer Balls". American
Scientist. 94 (4):
350. doi:10.1511/2006.60.350.
3 Wikipedia
4 "The History of the Soccer Ball Part 2". Soccer Football World. Rig-Tech Inc. 9 June 2006. Archived from the original on 19 June 2010. Retrieved 29 September 2019.
5 ^ "World's First Intelligent Soccer Ball Receives FIFA Recognition". Cision. PR Newswire. 6 July 2012. Archived from the original on 23 July 2015. Retrieved 21 July 2015.
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