A parallelogram mechanism is used when an object needs to move up, down or sideways while remaining level. The rule of opposite sides being parallel keeps the things moving in a straight line. Drawing tools and lifting devices are built on this principle.
A Pantograph is a tool used by architects and map-makers to copy a drawing at a larger or smaller scale. It is made of four rods forming a parallelogram. As the side lengths and parallelism of a parallelogram cannot be changed, when we move one vertex to trace a drawing, another vertex moves in the exact same pattern but at a different scale.
In factory robot arms, parallelogram mechanism is used when the operating part must move while keeping the angle same. If a robot is lifting a component from a conveyor belt and placing it into a machine, the object should not tilt during motion. In a parallelogram linkage, if the top bar tilts or moves, the opposite bar follows in such a way that the shape remains a parallelogram. This way the part or object can be moved without unwanted rotation.
The hexagon is often called one of nature’s most efficient shapes. While squares and triangles may look simpler, the hexagon offers a special mathematical balance.
In drone engineering, the hexagon is a preferred design choice. Many drones meant for heavy lifting use a hexagonal layout. A square layout can work but a hexagonal layout gives better balance and control. With square layout, four motors may not be enough because each motor has to carry a larger share of the load. A hexagonal layout places six motors evenly around the centre with 60∘ separation between arms. It provides rotational symmetry which means that the drone is perfectly balanced in every direction and thrust is distributed more evenly. This polygonal arrangement ensures that the thrust from each motor is distributed equally. A drone can thus hover in a stable manner and manoeuvre better. The hexagonal shape is also more efficient and practical than an octagonal shape as the latter adds to more weight with lesser spacing between arms i.e. 45∘.
Hexagonal symmetry is used inside the materials of airplanes and spacecraft. In these honeycomb panels used for building aircrafts, a lightweight core made of hexagonal cells is placed between thin outer sheets. A hexagonal grid uses the least amount of material to create a lattice that is resistant to compression and bending. Just like honeycombs are hexagonal to store the most honey with the least wax, aerospace engineers use this shape to create rigid structures. With this, extra mass is avoided which is crucial for fuel efficiency.
In Civil and Structural Engineering, polygon geometry is the foundation. Buildings, bridges, roofs, floor plans, trusses, windows, tiles, roads and city layouts are all designed using polygonal shapes. Polygons are the industry standard for three specific reasons.
Space Optimization (Tessellation) – A tessellation is, by definition, a pattern of shapes that fits together with no gaps and no overlaps. In architecture, tessellations are useful when architects want to divide a large surface or space into repeated modules. These modules may be floor tiles, wall panels, ceiling patterns, window panels, paving blocks etc. As the shapes fit together perfectly, material is not wasted and there is efficient use of available space.
Load Distribution – Triangulating polygons is a powerful safety principle for civil engineers. Any quadrilateral can be split into two triangles by drawing a diagonal. This is important because a quadrilateral can easily change shape when force is applied. A rectangle made of four beams is naturally weak. On pushing it from the side, it turns into a parallelogram. This sideways deformation is caused by shear forces. By adding a diagonal beam, two non-deformable triangles are created. A concrete floor slab is supported by beams on its edges. When people, furniture or walls put weight on the slab, internal forces act inside the slab. In a square room, the load can be considered as divided into four triangles. The weight from each triangular zone is transferred towards the nearest supporting beam on the edge because the beam is the stronger supporting line. From the beams it finally travels down to the columns at the corners.
Aesthetics & Wind Resistance – Many modern skyscrapers (like the Burj Khalifa) are not simple square or rectangular towers. They are complex polygonal shapes that change as they get higher. With a polygonal cross-section, the building becomes more aerodynamic. By changing the face, width, direction and edges, the wind can never settle into a rhythmic push and without a rhythm, it cannot make the building resonate. Wind hits a flat square surface with full force. But when wind hits a polygonal stepped design, the air is broken up into smaller swirls of air called eddies. This prevents the skyscraper from swaying too much in the wind.
Geometry naturally emerges in nature to solve practical problems like how to store more material and survive with minimum energy. Nature always seeks the path of least resistance, using the minimum energy to get the maximum result.
Honeybees need a place to store their honey and raise their young. They have two requirements. They need to store as much honey as possible. Secondly, making wax is expensive as bees have to eat a lot of honey to produce a tiny bit of wax.
Circles would leave gaps between them (wasted space). Squares or Triangles would fit together with no gaps but would require more wall material (wax) to hold the same amount of honey. The hexagon encloses the most area for a given perimeter which means it needs the least wall length to hold a given amount of honey. When this is extended to a whole sheet of equal cells, it is known as the Honeycomb Conjecture. Bees don’t consciously build hexagons. They build roughly circular cells and the soft wax flows and settles into hexagons because that’s the lowest energy arrangement when equal cells press together.
In the Giant’s Causeway in Ireland, there are thousands of basalt columns that look like they were carved by hand into perfect hexagons. This is actually a physics phenomenon called Columnar Jointing. When thick lava cools, it shrinks and therefore cracks. Nature wants to release the tension of the cooling rock in the most efficient way possible. The lava naturally chooses the 120° angle because it is the most stable way to dissipate energy resulting in a field of hexagonal pillars.
Bubbles want the smallest possible surface area to hold their air. When multiple bubbles push against each other, they naturally rearrange themselves into a hexagonal lattice because it is the lowest energy state.
When we visit a famous cathedral or a historic parliament building, we see that the central hall is often an Octagon. An octagon is the closest a building can get to a circle while still using straight and easy-to-build walls. It gives a 360-degree feel making it perfect for theatres or courtrooms where everyone needs to see the centre.
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