DarkRange55
I am Skynet
- Oct 15, 2023
- 1,846
If triangles are the "strongest shape in nature" (or is that spheres because of the arc? I've also recently read that hexagons are strongest shape which is why bee's build honeycomb in hexagons for the surface tension strength or something along those lines.) So why aren't there more triangular designs found in nature? And is a triangle actually "the strongest shape in general?" If its a hexagon, why don't we build hexagonal or triangular tunnels? As one friend out it, (materials engineer), "I can tell you that although perhaps not in the visible or even microscopic realm, triangles play a pretty big role in how different materials gain their internal strength (ie. the toughness of metal). In common metal atomic lattices (crystal micro-structure), there are three basic arrangements (face-center cubic, body-center cubic and hexagonal close-packed to name some common ones). They have a variety of three dimensional arrangements but all of them have base atomic layers of either 'squares' or hexagons. In both cases it is how the next layer of atoms fixes itself to the base layer that plays a role in strength. In the case of the cubic structures one way is for the atoms to arrange themselves centered in the 'squares' of the base layer which forms a series of four sided pyramids. This atomic packing style continues and we end up with a series of diamond-like structures. For hexagonal close-packed it gets considerably more complex but triangles can still be involved."
A cylinder would be stronger for supporting a given force, but a pyramid would be stronger for supporting a given pressure (the small tip would not have a lot of force even at high pressure).
A corner of a triangle is easier attacked from the two exterior sides, i.e. there are three weaker regions in the triangle (compared to the central region). The way triangles distribute weight when they are in a group makes them stronger. A single arch is stronger, but when you use lots of triangles when building a bridge it becomes stronger than using one arch. That is why we use triangles for most of our construction. In the circle there are no weaker regions and the limiting perimeter is minimal irt the protected area/volume.
Hexagonal cylinders are the ideal shape for things like the larvae because they maximize the usable volume for longer, round objects while minimizing the materials for the walls. True round cylinders are better at resisting compressive forces, but a beehive is not under much compression. Triangles are great at resisting point compression
The triangle isn't a "strong" shape. It's just that if you form a triangle with sides of fixed lengths, they can only make a single possible shape. Not so with a quadrilateral, which has multiple degrees of freedom for rotation at each hinge. This makes the triangle stable, but not strong. In three dimensions, stability is still achieved with triangles, just more of them and in more planes. Euclid's third criterion for triangle congruency is that two triangles with three equal sides are congruent. This is only true for triangles, not for shapes with more than three sides. For example you can build infinite rhombi (one of which will be a square) whose sides have a given length, but there is only one equilateral triangle whose sides have a given length. Durability from elemental damage is a function of thickness and nothing else. A sphere would take the least amount of damage per unit volume, but I'm not sure what value that particular metric has. A sphere also has the most volume to surface area of any shape. Which I would guess means it effectively has the highest thickness per unit of surface area. Maximum volume. Minimum surface area. That is why rocks approach round when they have been ground by ice a mile thick 12000 years ago. But the sphere is fairly hard to make comparatively speaking so if you aren't able to make a sphere with the tools at hand the closest thing to a sphere would be best right like a cube would be stronger than a pyramid? If you build a rectangle out of four pieces of wood and four screws, you can deform it into a parallelogram by applying a force on the corners. If you tighten the screws you'll need to apply a stronger force but it will still be possible to deform it. This is because the length of the four sides are not sufficient to identify the four angles of the shape. If you do the same for a triangle, you will not be able to deform the shape except by tearing it apart (removing the screws) or modifying the length of the sides. You can modify the length of the sides by e.g. tension or compression, but of course the required force is much stronger than in the case of a rectangle. Because you can't change the angles without change the lengths of the sides. With a square, it can deform into a rhombus without bending or otherwise changing the lengths of the sides. With an equilateral triangle, you have to do both.
Most durable shape: I think it would be a sphere because it has the minimum possible stress points but I'm not sure: Highest possible contact stresses.
This is a complicated question, because "durability" is vague. My knowledge of material properties would say that the material plays a larger role than the shape persay. "The problem was just purely for the shape right like ignoring the material assuming that you have the best material possible." Sure but even then, the crystalline structure of the material will dictate how forces are expressed through the shape. A FCC vs a BCC vs a HCP material shape will be have different strengths and weaknesses along different axis.
I think you're missing the point, which is that material selection is incredibly important, and you can't just say "the best material", because that's not really how engineering works. Do you want to avoid corrosion? Is surface hardness important? What about heat conduction? Abrasion resistance, and if so with what other materials? A cube of salt will be less durable if you chuck it into a lake than a cube of steel. A cube of steel will be less durable if you chuck it into an ocean than a cylinder of aluminum. A cylinder of aluminum will be less durable in a volcano than a sphere of quartz.
There is no one best shape, the whole point of mechanical engineering is finding the "best shape", though with more useful constraints applied than "most durable". Depending on if you mean fatigue or abrasion resistance (or impact toughness in the most conventional sense) there are different materials that outperform each other in all of those categories.
The cylinder can support the most weight—in this case—because its walls don't have any corners. This means the force from above is evenly shared across the cylinder. The square and triangle columns carry the weight on their corner spots. These corners are weak and these columns collapse because the weight at their edges is too strong. You can see this happen as they begin to twist and bend right before they fall. Columns are used in all types of architectural designs, both to support structures and for their beauty.
If you want to contain pressure, such as a tank of compressed air, the circle is far better than the triangle. That holds true for external pressure as well, such as a submarine or a deep ocean explorer. But if you want to build a tower, or a long straight bridge, then it should be built out of many triangles.
The pyramid (tetrahedron) is useful only for vertical load imposed at its peak. If you place a horizontal load at the top of the pyramid, it will easily shear off. We refer to load coming from all directions as a hydrostatic load. This is the load a submarine or a pressure tank will feel. A pyramid shape would be terrible for hydrostatic load. The corners and edges will act as pressure points, dramatically increasing the local stress and causing failure. There's a reason all submarines are spheres or cylinders.
A pyramid is useful for two cases. One, it is the strongest shape for a free-standing solid structure. Think the pyramids of Giza here. Secondly, it is useful for spreading out load, say via a footing. Pyramidal footings are sometimes used in buildings. A large steel column will pierce right through the ground, so you put a concrete pedestal, sometimes a pyramid, to spread out the load.
Spheres, they're good for hydrostatic pressure, whether internal or external. If you need to dive deep into the ocean or hold a high-pressure gas, spheres are your friend.
It is important to clarify that it really depends on the situation that each shape is in. Material matters more than shape. i.e, I read that triangles are stronger but, concrete can't handle forces in that way.
Triangles and pyramids are frequently used because they split the force applied on them into tension/compression forces applied onto their beam members. (Statics) This is why bridges frequently have triangular truss members... because the force vectors placed on them are largely directional and not isotropic (they come from specific directions, not from all directions equally).
So in instances where the force applied to your shape is highly directional/anisotropic, a triangular member is usually superior... it splits the force into components that are placed along each beam in the triangle structure. This is highly simplified, and of course it depends on the angles of the triangle, the thickness of the beams, the material of the beams, the magnitude of the force, etc… but basically, the triangle is a great building block for structures.
The sphere is great when the force is isotropic in all directions... like in a pressure chamber. This is because the pressure (just the force divided by the area) is so low along the surface of the sphere because the AREA of the sphere is maximized for the solid shape. If you start to make a sphere have dimpled or caved surfaces, you might increase the area... but you also mess with the statics of the solid sphere (you basically break up the forces into anisotropic, directional components... which is not the reason you use a sphere! ALTHOUGH: see geodesic domes).
So where would you use a sphere over a triangle? There are not many instances (that I can think of) where you would. Pressure differential forces (due to a difference in pressure...think the inside of a submarine versus the outside ocean, or the inside of a spaceship vs the outside vacuum of space, or even a blown up balloon) would like spheres. This is why the viewing glass in many spaceships or deep ocean robots are usually spherical. But humans encounter directional forces much more often than these omnidirectional forces, so the sphere is not often used.
Hence the triangle is generally regard as "the strongest shape in nature," which may on fact be somewhat of a misnomer.
Why doesn't this show up more in nature? But if I had to take a guess, I'd say that it's because it would require a kind of "loop" in body structure that could be very hard to accomplish simply through evolution (i.e. a tree with a branch that loops back to the trunk). As for non-organic objects, are there any examples that would necessitate that kind of triangular formation? I think it is encountered in nature, just on a much smaller scale. Structural proteins that compose scaffold-like structures commonly form triangles. The tetrahedron, (I think) is ubiquitous in nature. It is a pyramid made up of 4 triangles. This shape is fundamental in organic chemistry and nature. This is the shape of a water molecule (counting free electron pairs), single-bonding carbon compounds (alkanes e.g. methane), and many, many, many more. An atom at the center of a tetrahedral molecule is sp3 hybridized. Look at the cross-section of a seashell, or any spiral. The cross section of a vulture's wing bone structure. All your muscles are connected in a triangle. 2 bones as two of the sides, and the muscle as the third side. Triangles also give their name to trigonometry, which allows us to deal with circles, rotations, waves, signal processing, complex numbers, exponentials, and much more.
Like a sphere, a circle is strong if you don't know which direction forces are going to come from, or forces come from all directions. But a circle also doesn't have strong spots - a dedicated shape can have strong spots oriented toward where the forces will be strongest if that is known in advance. A square, or a brick, is the strongest if the force is applied evenly across one direction.
The strength of the individual bonds has nothing to do with the geometric strength of triangles and pyramids, but the stiffness of the bonded material does. Diamond is thus very strong because it has strong bonds arranged in a strong shape, but its crystalline lattice makes it vulnerable to cleavage along certain planes. Graphite is very strong in two dimensions because it has very strong bonds in a fairly strong shape, but in the third dimension it has a weak bond in a weak shape.
The nucleus of an atom has a shape, and not all are spherical. However I know of no way for humans to "build" a shape smaller than an atom. (See https://en.wikipedia.org/wiki/Femtotechnology)
Stability is part of the strength of the shape. But note that a square is as stable as a triangle if it is not hollow.
Rocks approach round (or ellipsoidal) because the points get bashed off first.
A sphere is easy to make if you don't mind endless polishing.a series only the stronger shape if you don't know which direction for applied force will come from, or if it comes from all directions. A pyramid is stronger if the force comes from a point, while a cube is stronger if the force comes from a direction and is spread over a plane.
For durable shapes, look at bearings. These come in spheres, cylinders, and tapered cylinders. But material is more important in terms of durability, as are the stresses to which it will be subjected.
A true rhombus, all sides equal, is not a very strong shape. But add one diagonal and it becomes very strong.
Squares are not a stable shape compared to triangles, and are not as efficient users of perimeter as hexagons or circles, so nature doesn't use them very often. However humanity's use of them is not arbitrary - rectangular buildings pack beautifully and are easy to put roofs on, and rectangles are highly suitable for mass-produced things such as bricks and boards.
Both material and shape are important,. For example, try building with tiny diamond spheres, or with beams made of cotton candy.
A cylinder would be stronger for supporting a given force, but a pyramid would be stronger for supporting a given pressure (the small tip would not have a lot of force even at high pressure).
A corner of a triangle is easier attacked from the two exterior sides, i.e. there are three weaker regions in the triangle (compared to the central region). The way triangles distribute weight when they are in a group makes them stronger. A single arch is stronger, but when you use lots of triangles when building a bridge it becomes stronger than using one arch. That is why we use triangles for most of our construction. In the circle there are no weaker regions and the limiting perimeter is minimal irt the protected area/volume.
Hexagonal cylinders are the ideal shape for things like the larvae because they maximize the usable volume for longer, round objects while minimizing the materials for the walls. True round cylinders are better at resisting compressive forces, but a beehive is not under much compression. Triangles are great at resisting point compression
The triangle isn't a "strong" shape. It's just that if you form a triangle with sides of fixed lengths, they can only make a single possible shape. Not so with a quadrilateral, which has multiple degrees of freedom for rotation at each hinge. This makes the triangle stable, but not strong. In three dimensions, stability is still achieved with triangles, just more of them and in more planes. Euclid's third criterion for triangle congruency is that two triangles with three equal sides are congruent. This is only true for triangles, not for shapes with more than three sides. For example you can build infinite rhombi (one of which will be a square) whose sides have a given length, but there is only one equilateral triangle whose sides have a given length. Durability from elemental damage is a function of thickness and nothing else. A sphere would take the least amount of damage per unit volume, but I'm not sure what value that particular metric has. A sphere also has the most volume to surface area of any shape. Which I would guess means it effectively has the highest thickness per unit of surface area. Maximum volume. Minimum surface area. That is why rocks approach round when they have been ground by ice a mile thick 12000 years ago. But the sphere is fairly hard to make comparatively speaking so if you aren't able to make a sphere with the tools at hand the closest thing to a sphere would be best right like a cube would be stronger than a pyramid? If you build a rectangle out of four pieces of wood and four screws, you can deform it into a parallelogram by applying a force on the corners. If you tighten the screws you'll need to apply a stronger force but it will still be possible to deform it. This is because the length of the four sides are not sufficient to identify the four angles of the shape. If you do the same for a triangle, you will not be able to deform the shape except by tearing it apart (removing the screws) or modifying the length of the sides. You can modify the length of the sides by e.g. tension or compression, but of course the required force is much stronger than in the case of a rectangle. Because you can't change the angles without change the lengths of the sides. With a square, it can deform into a rhombus without bending or otherwise changing the lengths of the sides. With an equilateral triangle, you have to do both.
Most durable shape: I think it would be a sphere because it has the minimum possible stress points but I'm not sure: Highest possible contact stresses.
This is a complicated question, because "durability" is vague. My knowledge of material properties would say that the material plays a larger role than the shape persay. "The problem was just purely for the shape right like ignoring the material assuming that you have the best material possible." Sure but even then, the crystalline structure of the material will dictate how forces are expressed through the shape. A FCC vs a BCC vs a HCP material shape will be have different strengths and weaknesses along different axis.
I think you're missing the point, which is that material selection is incredibly important, and you can't just say "the best material", because that's not really how engineering works. Do you want to avoid corrosion? Is surface hardness important? What about heat conduction? Abrasion resistance, and if so with what other materials? A cube of salt will be less durable if you chuck it into a lake than a cube of steel. A cube of steel will be less durable if you chuck it into an ocean than a cylinder of aluminum. A cylinder of aluminum will be less durable in a volcano than a sphere of quartz.
There is no one best shape, the whole point of mechanical engineering is finding the "best shape", though with more useful constraints applied than "most durable". Depending on if you mean fatigue or abrasion resistance (or impact toughness in the most conventional sense) there are different materials that outperform each other in all of those categories.
The cylinder can support the most weight—in this case—because its walls don't have any corners. This means the force from above is evenly shared across the cylinder. The square and triangle columns carry the weight on their corner spots. These corners are weak and these columns collapse because the weight at their edges is too strong. You can see this happen as they begin to twist and bend right before they fall. Columns are used in all types of architectural designs, both to support structures and for their beauty.
If you want to contain pressure, such as a tank of compressed air, the circle is far better than the triangle. That holds true for external pressure as well, such as a submarine or a deep ocean explorer. But if you want to build a tower, or a long straight bridge, then it should be built out of many triangles.
The pyramid (tetrahedron) is useful only for vertical load imposed at its peak. If you place a horizontal load at the top of the pyramid, it will easily shear off. We refer to load coming from all directions as a hydrostatic load. This is the load a submarine or a pressure tank will feel. A pyramid shape would be terrible for hydrostatic load. The corners and edges will act as pressure points, dramatically increasing the local stress and causing failure. There's a reason all submarines are spheres or cylinders.
A pyramid is useful for two cases. One, it is the strongest shape for a free-standing solid structure. Think the pyramids of Giza here. Secondly, it is useful for spreading out load, say via a footing. Pyramidal footings are sometimes used in buildings. A large steel column will pierce right through the ground, so you put a concrete pedestal, sometimes a pyramid, to spread out the load.
Spheres, they're good for hydrostatic pressure, whether internal or external. If you need to dive deep into the ocean or hold a high-pressure gas, spheres are your friend.
It is important to clarify that it really depends on the situation that each shape is in. Material matters more than shape. i.e, I read that triangles are stronger but, concrete can't handle forces in that way.
Triangles and pyramids are frequently used because they split the force applied on them into tension/compression forces applied onto their beam members. (Statics) This is why bridges frequently have triangular truss members... because the force vectors placed on them are largely directional and not isotropic (they come from specific directions, not from all directions equally).
So in instances where the force applied to your shape is highly directional/anisotropic, a triangular member is usually superior... it splits the force into components that are placed along each beam in the triangle structure. This is highly simplified, and of course it depends on the angles of the triangle, the thickness of the beams, the material of the beams, the magnitude of the force, etc… but basically, the triangle is a great building block for structures.
The sphere is great when the force is isotropic in all directions... like in a pressure chamber. This is because the pressure (just the force divided by the area) is so low along the surface of the sphere because the AREA of the sphere is maximized for the solid shape. If you start to make a sphere have dimpled or caved surfaces, you might increase the area... but you also mess with the statics of the solid sphere (you basically break up the forces into anisotropic, directional components... which is not the reason you use a sphere! ALTHOUGH: see geodesic domes).
So where would you use a sphere over a triangle? There are not many instances (that I can think of) where you would. Pressure differential forces (due to a difference in pressure...think the inside of a submarine versus the outside ocean, or the inside of a spaceship vs the outside vacuum of space, or even a blown up balloon) would like spheres. This is why the viewing glass in many spaceships or deep ocean robots are usually spherical. But humans encounter directional forces much more often than these omnidirectional forces, so the sphere is not often used.
Hence the triangle is generally regard as "the strongest shape in nature," which may on fact be somewhat of a misnomer.
Why doesn't this show up more in nature? But if I had to take a guess, I'd say that it's because it would require a kind of "loop" in body structure that could be very hard to accomplish simply through evolution (i.e. a tree with a branch that loops back to the trunk). As for non-organic objects, are there any examples that would necessitate that kind of triangular formation? I think it is encountered in nature, just on a much smaller scale. Structural proteins that compose scaffold-like structures commonly form triangles. The tetrahedron, (I think) is ubiquitous in nature. It is a pyramid made up of 4 triangles. This shape is fundamental in organic chemistry and nature. This is the shape of a water molecule (counting free electron pairs), single-bonding carbon compounds (alkanes e.g. methane), and many, many, many more. An atom at the center of a tetrahedral molecule is sp3 hybridized. Look at the cross-section of a seashell, or any spiral. The cross section of a vulture's wing bone structure. All your muscles are connected in a triangle. 2 bones as two of the sides, and the muscle as the third side. Triangles also give their name to trigonometry, which allows us to deal with circles, rotations, waves, signal processing, complex numbers, exponentials, and much more.
Like a sphere, a circle is strong if you don't know which direction forces are going to come from, or forces come from all directions. But a circle also doesn't have strong spots - a dedicated shape can have strong spots oriented toward where the forces will be strongest if that is known in advance. A square, or a brick, is the strongest if the force is applied evenly across one direction.
The strength of the individual bonds has nothing to do with the geometric strength of triangles and pyramids, but the stiffness of the bonded material does. Diamond is thus very strong because it has strong bonds arranged in a strong shape, but its crystalline lattice makes it vulnerable to cleavage along certain planes. Graphite is very strong in two dimensions because it has very strong bonds in a fairly strong shape, but in the third dimension it has a weak bond in a weak shape.
The nucleus of an atom has a shape, and not all are spherical. However I know of no way for humans to "build" a shape smaller than an atom. (See https://en.wikipedia.org/wiki/Femtotechnology)
Stability is part of the strength of the shape. But note that a square is as stable as a triangle if it is not hollow.
Rocks approach round (or ellipsoidal) because the points get bashed off first.
A sphere is easy to make if you don't mind endless polishing.a series only the stronger shape if you don't know which direction for applied force will come from, or if it comes from all directions. A pyramid is stronger if the force comes from a point, while a cube is stronger if the force comes from a direction and is spread over a plane.
For durable shapes, look at bearings. These come in spheres, cylinders, and tapered cylinders. But material is more important in terms of durability, as are the stresses to which it will be subjected.
A true rhombus, all sides equal, is not a very strong shape. But add one diagonal and it becomes very strong.
Squares are not a stable shape compared to triangles, and are not as efficient users of perimeter as hexagons or circles, so nature doesn't use them very often. However humanity's use of them is not arbitrary - rectangular buildings pack beautifully and are easy to put roofs on, and rectangles are highly suitable for mass-produced things such as bricks and boards.
Both material and shape are important,. For example, try building with tiny diamond spheres, or with beams made of cotton candy.
