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The ‘Spherical’ Earth

In this blog post we will talk about the ‘spherical’ earth that we are all living in. Some people have a common misconception that the earth is flat, and for millennia this has been a carefully discussed topic especially in the middle ages where religious beliefs ruled the entire society.

We will try and debunk utilizing state-of-the-art photos, mathematics for describing our ’round’ earth and finally come with some facts about the earth and common misconceptions which people should be careful not to fall against.

Is the earth flat?

Before starting off on our talk about our beloved earth lets try and answer one of the simplest questions most people have asked themselves: Is the earth flat? To start off with, it certainly seems flat to the naked eye, as the horizon on which you can steer out into creates a seemingly straight line as expected with a flat earth. One interesting video about the flat earth is the ‘Vsauce’ video titled: “Is Earth actually flat?” A contribute to debunking the flat earth society.

Video 1: Vsauce video about the flat earth society and whether or not the earth is a flat disc.

For centuries this fact has puzzled human kind who believed to be the center of the universe and the earth to be a flat plate swimming alongside the back of a turtle, or in some cases, a flat disc where the sun has been placed on a curved space with a stationary earth.

Fighting the ‘Globe’ Theory

One of the latest’s and most famous examples of such a phenomena is the 1893 drawing by Richard Ferguson, a firm believer in the flat earth theory constructed the ‘plausible’ drawing of how the earth might be a curved square space with a stationary centered earth against the entire solar system and how this revolved around the earth system. He constructed this in order to ‘combat’ the dominant ‘Globe’ theory.

The 'square and stationary earth' theory proposed by Orlando Ferguson meant as a fight against the dominant 'Globe Theory'
Figure 1: The flat earth fighter Orlando Ferguson in 1893 fighting against the ‘Globe Theory’ the square space contains numerous biblical passages which is used to justify the wrong assumptions.

Even to this day with our photographic evidence from satellites and space missions, there exists people who remain members of the so-called flat-earth society, a society of firm believers in the flat earth theories that exist.

Photographic evidence of ‘Globe’ earth

After the invention of satellites in the 20’th century we now have photographic evidence of the so-called ’round’ earth as can be seen in Gallery 1. Here a collection of pictures of our blue ’round’ earth have been taken showcasing clearly that the earth is round.

This earth looks a lot like a sphere and surely this is a way better approximation of the shape of the earth rather than the flat earth society experienced. The following shows how to calculate the spherical earths volume, circumference and surface area based on a supplied radius of the earth.

Related read: Top ten misconceptions about ChatGPT, Top ten misconceptions about Earthquakes, Top ten misconceptions about Tsunamis

The Spherical earth

Now lets dive down into the mathematical description of our spherical earth. A sphere is most easily described by its radius, r, from which you can calculate the volume, V, and surface area, A, along with the two-dimensional circumference S.

(1)   \begin{align*} V &= 4/3 \cdot \pi \cdot r^3 \\ A &= 4 \cdot \pi \cdot r^2 \\ S &= 2 \cdot \pi \cdot r \end{align*}

For the determination of the radius of the earth there exist multiple values based on a variety of methods a great overview over the different methods is given in a Wikipedia article:

The most common values ranges between 6.384 km – 6.528 km for a radius of the earth. This quite different range details the differences in roundness that the earth resembles, as the mountains and sea-floor valleys aren’t accurately mapped in a perfect spherical world.

The non-spherical earth

The spherical earth which we live on is therefore not exactly round as first expected as the mountains valleys and later realized poles of the earth aren’t accurately mapped in a spherical system. The

In fact when mathematicians worked on theories of spherical earths and tidal forces, wind speed generations, Coriolis forces to mention a few. These are facts about the earth which a full perfect sphere cannot perfectly answer.

The slight variations in shape and size of the earth have significant impacts on the way our planet behaves. For example, the mountains and valleys affect wind patterns and ocean currents, and the non-uniform distribution of mass leads to variations in gravity, causing fluctuations in the Earth’s rotation.

Despite these complexities, the spherical model of the earth has proven to be incredibly useful for navigation, satellite communication, and understanding the world around us. But it’s important to recognize that the earth is not a perfect sphere and that its imperfections have real-world effects.

Mathematical description of imperfections

The description of the deviations from the spherical shape is mathematically described through the flattened ellipsoid equations where an ellipsoidal disc can be described simply through two different equations concerning the radii typically described in the equations of state.

The equations used to describe the surface area, volume and in a similar format as that of a spheroid except for the fact that the parameters; a, b and c are introduced instead of the value of r.

Ellipsoids for describing the Globe type earth which is more accurate than the spheroid version.
Figure 2: Illustration of the three typical ellipsoids with different values of the radii parameters a, b and c. For the special case where a=b=c we have a spheroid. See source.

(2)   \begin{equation*} (\fraq{x}{a})^2 +(\fraq{x}{b})^2 + (\fraq{x}{c})^2 = 1 \end{equation*}

Now for the cases where surface areas of ellipsoids need to be calculated the expressions are quite a bit more complicated as the surface integrals depend on the chosen parameterization of the different values of a, b and c among others.

The parameterization typically found used in spherical projections correspond to spherical coordinate systems utilized in many different aspects of life and engineering in general. The parameterization follows to be the following:

Volume of an ellipsoid:

(3)   \begin{equation*} V = \frac{4}{3} \cdot \pi \cdot a \cdot b \cdot c \end{equation*}

While the general formula for spherical coordinates and therefore parameterized surface area of the ellipsoid is the following:

(4)   \begin{equation*} \begin{bmatrix}   x = a \sin(\delta) \cos(\theta) \\   y = b \sin(\delta) \sin(\theta)\\   z = c \cos(\theta) \end{bmatrix}  =  R \cdot  \begin{bmatrix}   \cos(y)\cos(\lambda) \\   \cos(y)\sin(\lambda) \\    \sin(y) \end{bmatrix} \end{equation*}

Where -1/2 \leq y \leq 1/2 and 0 \leq \lambda \leq 2 \pi. The definition of R is described as

(5)   \begin{equation*} R = \frac{abc}{\sqrt{c^2 (b^2 \cos^2 \lambda + a^2 \sin^2 \lambda) \cos^2 y + a^2 b ^2 \sin^2 y}}, \quad -\pi/2 \leq y \leq \pi/2, \quad 0 \leq \lambda \leq 2 \pi \end{equation*}

These equations are used in various applications from satellite tracking systems to navigation systems for vessels and ship, to map creation and transformations utilized for creating specific map types.

Now that we have established that the earth is round one may consider approving of this theory and buying a globe to support the ‘globe’ theory. This helps the website and content creator if you decide to click and buy one of the recommended globe spheres below.

Natural materials

Natural materials – silt

In this blog post we will characterize the natural material silt. We will try and answer some of the most important questions in relation to silts such as; What is silt? What is silt used for? What is the difference between silt, sand and clay? Does silt pollute water? What is siltation?

Definition of silt

Silt is a characteristic soil whose grain size lies between that of clay and sand with sizes ranging between 0.002 mm and 0.063 mm.

Related read: Soil characteristics – sand, Natural materials – clay

Silt is a detritus material meaning that it stems from weathered rock material and you can actually feel the individual particles between your fingers if rubbed gently. The physical properties of silt is hard to make general characterizations about as the individual properties depend on its ability to adhere to water and drain. An example of possible explanatory properties is the Atterberg limits.

Atterberg limits

The silty soils adherence to water makes is hard to accurately characterize. This means that the defining characteristic when it comes to soil dynamics is the ability to describe the Atterberg limits of silty soil. Atterberg limits are used to describe cohesive materials for their strength and water limit, see source.

The Atterberg limits are a simple way of describing the bearing capacities of wetted silt and clayey soils. These Atterberg limits are some of the defining limit states behind the soil characteristics and are used to accurately describe the water content and corresponding soil behavior in a simple manner.

Individual soil states

The Atterberg limits are described in four different manners called states. These states are given from silty and clayey soils and includes; Solid, Semi solid, Plastic and Liquid.

The definition of each of the subzones varies based on the type of clay and silt being investigated. Here the defining property is the ability to adhere and keep water content within the soil. Generally speaking, as the silty soils water content increases, it transforms its characteristics from that of a solid towards a liquid.

Atterberg limits visualized

The key defining characteristic behind the definitive behavior of silts and clays is their ability to keep water within the substrate. This is best illustrated from the resulting stress-strain curves usually employed when defining solid state materials bearing strengths. An illustration of the different characteristic states of the material alongside the bearing capacity of each state is outlined in Figure 1.

Illustration of the Atterberg limits where the individual soil characteristics are outlined this includes differences within the Solid Limit, Plastic Limit and Liquid Limit state space of the watery soils
Figure 1: Illustration of Atterberg limits with different strength characteristics shown versus different states of the material. This includes the stress-strain curves of the material which describe the material properties through the SL, PL and LL states

With this initial explanation of the Atterberg limits, it quickly becomes evident the importance of defining each of these limits in an accurate manner.

Thought process behind silty\clayey soils bearing capacity

To set the stage for the upcoming explanation of why these limits are important, lets try and imagine a scenario where a large infrastructure project is trying to be built.

Firstly one may investigate the initial area visually ensuring that there are no immediate obstacles needing to be removed before geotechnical investigations might be performed.

Second, a set of initial tests have examined the cross-section of the earth and found out that the initial state of the underground consists of a mixture of clay and silty soil.

Third, now one may wonder what the initial strength of such soils might be and if at all the soil exhibits sufficient capacity for bearing the load of the structure proposed for the site in question.

In conclusion, it becomes evident that the bearing capacity of silty clayey soils must be understood properly for the construction of larger infrastructures to be feasible.

As we now understand, it is important to estimate the limiting behavior of soils when subjected to loads and therefore we need to calculate the Atterberg limits. In the following sections we dive deeper into the calculation of the different types of Atterberg limits. Starting of with the liquid limit, LL.

The Casagrande cup experiment

The Casagrande cup experiment is used for the determination of the soil water content based on a standardized number of blows until a small valley created through use of specialized instruments collapses. See Figure 2 for an illustration of the equipment and resulting collapse of the valley.

The Casagrande test experiment illustration of the equipment utilized alongside with an explanatory figure of a silty soil sample.
Figure 2: Casagrande cup filled with substrate. Top left: valley and filled cup before number of blows. Bottom left: Collapsed valley after x number of blows. Right: Illustration describing the specialized equipment used for creating the valley and material used as shock absorption.

When the valley starts to fall together the number of blows is numbered on a piece of paper. The soil sample is then weighed and dried with the resulting weight before and after measured carefully. This way you can determine the number of blows before failure mechanisms of the substrate are activated as a function of the water content of the sample.

A relationship between the number of blows and the water content can then be drawn and theoretical distributions for the soil strength can be inferred through careful regression techniques. An example of a given relationship between water content and number of blows is outlined in Figure 3. The predetermined liquid limit is theoretically defined for 25 number of blows to the sample.

Describing the water content and corresponding number of blows in the Casagrande test for identification of the liquified limit of the soil sample.
Figure 3: Water content versus number of blows within the Casagrande cup experiment before the collapse of the shear walls. The liquid limit of the Casagrande experiment corresponds to the straight line interception at 25 blows.

The liquid limit described by this test allows for an easy and fast interpretation of the liquid limit of silty clayey soils.

Now in order to try and estimate the plastic limit of the soil we need another test measure technique called the rolling test.

Rolling tests determining plasticity level

In order to determine the plastic limit of the soil, the limit where constant stress cause constant strain, we need to perform a rolling test on a sample of the substrate.

The methodology of performing a rolling test is in short to roll out a specific sample of the soil under assumptions of constant stress and cross section until the water content start to crack. When cracking is initiated the sample is quickly weighed before being dried out in the lab heaters for 24 hours.

Soil sample roll speed

When rolling the soil sample out you are actually drying out the sample therefore it is important to realize that the speed at which you are rolling out the sample influences the drying rate and correspondingly can result in errors as the sample is considered to contain a constant water content.

Faster speeds and pressure on the silty soil sample will cause inaccurate measurements of the water contents due to the pressure distribution on the soil pushing out water from underneath the soil sample.

A characterization technique for the natural material silt.
Figure 4: Example of a rolled silt soil experiment. The silty soil is observed through use of the cracked and jagged sample

However, rolling out the soil sample slowly is neither a good option as the contents of the water sample is likewise going to dry out before being able to accurately predict the water contents.

A perfect roll out speed is necessary for the plastic limit of the soil to be accurately described. This speed depends and varies individually from the different soil samples and thus expert engineering judgement is needed for determination of the plastic limits.

Differentiating clay from silt

When trying to investigate soil properties and before determining and characterizing the difference qualitative parameters of the soil sample it is important to consider the distinctive behavior’s separating silt from clay. The distinction is easily made through use of two different test techniques. Firstly the ‘Elephant’ test technique is easily utilized

Elephant test technique

The elephant test technique for distinguishing soil behavior’s is quickly and accurately utilized even without expensive field laboratory equipment. The distinctive test is performed by creating an elephant like trunk shape of a small soil sample which is approximately 3 mm thick in shape.

The shape of the constructed elephant trunk is then examined and seen whether or not it is able to withstand the force underneath its own gravity. If the trunk easily upheld its own shape under its own weight, then the soil sample is considered to be clay.

However if the soil sample cannot easily withstand the force of gravity with deformed shapes and cracks on the surface then the soil sample is considered silt. This is an easy and inexpensive way of making sure that the soil sample in question is actually silt and not clay and visa versa.

Another commonly used methodology is to create a bended U-shaped sausage which is then utilized to check for cracks in a similar manner as the elephant. These inexpensive ways of determining whether or not a soil sample is consisting of clay or silt is invaluable.

With this being said, there also exist other methodologies to easily investigate whether a soil sample is clay or silt one of these is called a sedimentation test.

Sedimentation testing

The sedimentation test is a technique similar to that of the elephant test in that this specific version is likewise meant as an example to try and differentiate the soil sample into clay and silt respectively. The clay particles when settling in a tube will create a mixture of solid particles and liquids called a suspension while a silt sample will create a coarser sample.

Suspension definition and behavior

A suspension is particular interesting when trying to investigate the hydrodynamic properties of a fluid and when trying to investigate eco-systemic impacts of large infrastructure projects.

The ecosystem can easily be affected by suspensions in the sense that endangered species are affected by floating clay particles where gills of fish are clotted and marine fauna such as eelgrass are threatened by sedimentation layers stealing available sunlight.

The problems also arrives when considering the time spent on the settling particle velocities which for a given suspension can be extremely small, allowing individual particles to easily stay in the suspended water column for extended periods of time before it will be able to settle on the ground surface.

For clay the particles can stay in suspension for hours even days without settling in still water due to the water undulations within the water column causing the fine material to keep sculpting around.

Settling behavior of silt

The difference in settling between clay and silt is easily measured when conducting sedimentation experiments. As explained, the clay settling velocities are small allowing for the suspension to keep particles in the water column for extended periods of time corresponding to days.

For silt the suspension will fall out of the water column much faster than clays and thus when investigating soil samples and characterizing the sample into clay and silt, measuring the settling time and velocities would allow you to easily distinguish the soil sample into clay and silt.

The silts settling velocity is much larger than clay due to mainly two phenomena, the particle sizes of silt are an order of magnitude larger and flocculation effects are often quite significant for silty materials. The flocculation is when particles of different sizes clump together during settling resulting in increasing settling velocity of the entire joint lump of particles compared to the individual ones.

This particulate behavior leads to a common phenomena called siltation.


Siltation is a phenomena common in river -streams and -mouths as a result of excavating, extreme rainfall events or pollution. The siltation of river streams poses environmental dangers to ecosystems as the consequent sedimentation strangles bottom feeding and living creatures, while simultaneously damaging eggs and larvae exposed to silted environments.

Some common example of siltation events occur during marine dredging where silted material is transported upwards in the water column. The upward momentum allows whirling of particles up into the undisturbed water column causing the individual particles to be transported large distances and spread out over potentially endangered areas.

Another example of siltation is during extreme rainfall events where soil banks are eroded away due to increased water volumes causing silt and clay rich banks to be eroded and washed into the water column.

When transported downstream and before settling down into more calmer waters this siltation of water streams can cause suffocation of fish and degradation of wildlife. It is therefore considered an environmental disaster when rivers are exposed to siltation events.

Natural materials

Natural materials – clay

In this blogpost we will explore one of the most common natural materials in the world, clay.

What is clay?

Clay is a natural fine-grained material which smoothens out easily under pressure and is often filled with water.

The composition of clay is typically filled with minerals such as kaolinite, etc. these minerals are important for the beauty industry as most common type beauty products utilize some form of clay minerals. Additionally clay minerals are important in certain types of pharmaceutical industries as binders, lubricants, diluents, pigments and opacifiers.

History of clay in pharmacy

The use of clay in the pharmaceutical industry dates back to prehistorian eras where it has been used in pottery for containers of medicine and in 1600 BC a ‘book’ containing evidence that clay was used against hemorrhages and other types of diseases has been found. Underlining the fact that clay is a crucial part of human society for thousands of years.!

Another prehistorian example of clay as a pharmaceutical instrument is In the 400’s BC a book by Hippocrates called “On Airs, Waters and Places”. In this book they describe the Armenian bole and clays usage in healing dysentery and diarrhea. The Armenian bole is a special type of clay often used as medicine, as a pigment and gliding material. It is often red due to the presence of iron oxide.

The important compounds of clay in pharmacy is their mineral composition. In order to characterize clay it usually is subdivided into four main groups corresponding to the mineral content. These are:

  • Kaolinite
  • Smecticte
  • Illite
  • Chlorite

Where further subdivisions can be made of clay based on the mineral plate layering as outlined in Table 1.

S. No.General FormulaGroupLayer Type
3Montmorillonate (Al1.67Mg0.33)
Si4O10(OH)2M + 0.33
Saponite:Mg3(Si3.67Al0.33)O10(OH)2M + 0.33
Palygoskite-sepiolite group
Table 1: Mineral compounds typically found in clay with subdivisions based on typical geometries and mineral composition.

Common use cases of clay

The usages of clay are plentiful and it has been used for various amounts of pottery throughout history. These types of pottery have played an important role when trying to establish a culture and society, as containers have been produced holding everything from water to spices.

In modern history after the industrialization it has been possible to mass produce pottery instead of individual people being employed full time with the occupation suddenly it was possible to free up time for individuals now focusing on something more productive such as innovation and rethinking new technologies.

Related read: Natural Materials – Sand, Common misconceptions of Tsunamis.

Clay pottery as food storage

Meanwhile clay pottery is still used today to store all kinds of inventory such as food, spices and vegetables. With the advent of vegetarianism it might be reasonable to ask: Is clay vegan friendly? And to that the answer is a sounding yes! Clay is absolutely vegan friendly as the compounds making up clay pottery is entirely from non-animal origins meaning that anything vegan that you put in a clay bowl or pottery stays vegan.

Additionally clay pottery is able to withstand high temperatures and doesn’t decompose easily when burnt. Furthermore the trash from clay pottery is entirely natural and thus no need to worry about environmental hazards from trashed clay pottery.

Clay as a building material

Another usage of clays is as a building material for bricks, huts and similar. Here the clay is used to mold bricks in huge ovens allowing for hardening. When hardened and burned, the resulting bricks are capable of withstanding outside weather, rain and storms.

Clay bricks for building have been used for millennia and such the technology of creating bricks from ‘mud’ is something that has withstood the test of time. In a primitive manner, the recipe for creating clay bricks is simple; Mix water, clay, straw and heat until hardened in high temperatures.

Creating bricks with bare hands

An excellent resource for showcasing the simplicity and ancient technology associated with the clay brick creation is the Youtube channel ‘Primitive Technology’. A guy shows in simple steps how to create clay bricks from raw materials.

Video 1: Fired clay bricks as a primitive technology for brick creation

This way of creating bricks is simple yet effective and can be understood through the use of ancient technology.

The flow of liquids through clay

Unlike other materials such as sand, clay has quite an adherence to water. it is hydrophilic in the sense that most clays adsorb water in huge amounts leading to saturated soils. Additionally water passes extremely slowly through clay meaning that it acts as plug stopping all of the water trying to go through.

This property of plugging water and other liquid substances from flowing through clay is important in industries such as oil and gas. In fact clay functions as a lid under which oil and gas collects in huge amounts in large oil reservoirs and often times act as a necessary predisposition for the existence of oil fields.

The drainage of oil wells is a complicated process where high pressures are utilized for pumping water and oil mixtures up towards the surface from deep underground aquifers. Here impermeable layers help build up necessary pressures through overburden stresses allowing the transformation from organic material toward hydrocarbon fossil fuels.

The overburden stressors and impermeable layers can lead to another phenomena called artesian wells.

Artesian wells

An important consideration when trying to identify potential locations for water extraction is the overburden pressures or internal fluid pressures in underground aquifers. These aquifers allow the flow of liquids over large distances and are typically separated by impermeable clay layers. The impermeable clay layers impose stresses on underlying layers resulting in increased internal water pressures. When trying to extract water from such configurations the resulting pressures in the liquid lead to the establishment of artesian wells see Figure 1-2.

An illustration of an artesian well, where the clay\impermeable layer is overlaying the aquifer layers.
Figure 1: Cross-section diagram of an artesian well, USA. Illustration published in Physical Geology by Mytton Maury (University Publishing Company, New York and New Orleans) in 1894. Digitally restored.
Location of the great artesian basin in Australia. The artesian basin is made of aquifers underlying impermeable layers such as clay.
Figure 2:The great artesian basin in Australia consisting of the largest artesian basin in the world spanning more than 1.7 million square kilometers.

The key defining characteristic between regular and artesian wells is the overburden stresses and internal fluid pressures. these allow for the easy extraction of liquids through since flow is automatic up to the surface meaning no need for pumping systems or otherwise extraction techniques.

Water in artesian wells

Artesian wells often contain water. This water is typically of high quality and easily drinkable as the aquifers functions as filtering material of harmful minerals and bacteria. Additionally water from artesian wells are often free from contaminant sources and thus often bottled as spring water more expensive than regular water from taps, examples of such include Fiji water among others.

Oil in artesian wells

In the beginning of the oil era, artesian wells played an important part for extraction of oil. The flowrate is large for aquifers allowing the extraction of huge amounts of oil and the artesian wells didn’t require expensive pumping instruments.

Additionally the locations of the artesian wells made extraction easily manageable as the land deposits at the time were full of oil. An example of such oil fields are displayed in GIF 1.

The land deposits were furthermore located in locations where large infrastructure existed making the pipeline laying easily manageable. This ensured that oil deposits kickstarted the production of oil in the United States within the first years of industrialization.

Clay for sculping

As a last addition clay is also fantastic for sculping sculptures. For the handy people around these sculptures are easily made utilizing a spinning wheel combined with sculpting clays such as monster sculpting clay.

The monster sculpting clay is exceptional for clay sculpting whether you are a beginner just trying it out for the first time or an experienced professional looking for a hard clay based mounding substrate to easily manipulate into beautiful structures or figures.


Mineral composition of clay, source.