Animal Architect - Bees' Choices

Why does the exterior of bee hives look like a structure formed by hexagonal cells, yet the base of each cell is formed by three regular rhombuses? The first question involves the classical Isoperimetric problem, which is to determine a plane figure of the largest possible area whose boundary has a specified length. Based on their instinct, the bees have chosen hexagon that allows them to build the honey combs in the most economical way. The second question is how to construct cells with the highest volume yet using the least amount of wax. Regarding the base of the cell, the bees have used three regular rhombuses to form the base of the hexagonal prism. The structure will form the smallest surface area given the fixed volume so that the least amount of materials is needed to build the honey comb. The astronomer, G. F. Maraldi, has measured the acute angle and obtuse angle of the cell's base, which came up to be 70°32" and 109°28". Compared to the results from calculation, which are 70°34" and 109°26", they only differ by 2". The natural world is really amazing!

Animal Mathematician - Spider Web

A spider web is a net-like structure created by a spider with the spider silk that it produces. It is used for habitation and catching preys. Different types of spiders build webs of different shapes and sizes. Orb webs are formed by spiral threads weaved around the spokes in a radial structure. Funnel webs are horizontal, sheet-like webs with a small funnel-like tube in the middle or off to a side of the web. Tangle webs have a shapeless jumble of threads on the upper half, whereas the lower half has suspending silk threads that touch the ground to capture preys. In sheet webs, the silk threads are intertwined in different directions. In general, the spokes in the spider webs are radii radiating out from the centre, while the spiral line are threads laid on the spokes and following the logarithmic spiral.

Did you know?

1. When the hawks flying in the sky hunt for small animals on the ground, they will usually find the best angle to ensure the fastest speed and highest accuracy.
2. When geckos are hunting for mosquitoes, flies, moths and other kinds of small insects, they usually crawl along a spiral line. In mathematics, this spiral line is called Spiral.
3. When the leafcutter bees use their jaws to snip circular or oval-shaped pieces from the leaves, the leaf pieces all come in the same shape and size as if they are cut by the same cutter mold.
4. Coral polyp treats its body as a calendar. Every year, 365 curvy lines are marked on its body, apparently one line per day.
5. When the red-crowned cranes are migrating, they usually flock together in the shape of the V-formation to flying in the sky. The angle of this shape is usually about 110°. Coincidently, half of this angle is also the angle in diamonds cube.
6. In winter, cats usually curl their bodies up into the shape of a sphere so that the surface area of their bodies exposed in the cold air is minimal. In this way, heat loss is also reduced to the minimum.

Interactive Games

Bee Hives

The bee hives built by bees are formed by hexagonal prisms. The open end of each cell is a hexagon, and the base at the closed end is formed by three rhombuses of the same size. The acute angel and obtuse angle of each rhombus are 70o32" and 109o28".

Leaf

Does the arrangement of leaves follow special numbers? If the growth of leaves follows a rotation of 137.5°, they can avoid the overlapping of leaves to the largest extent, hence absorbing sunlight more efficiently.

Butterfly

The wings of butterflies are symmetrical but the frequency of wings flapping is not high, only four to ten times per second. Therefore, their symmetrical wings actually are not very useful in helping them maintain a balanced flight.

Starfish

Starfish is of the shape of a star and normally has five arms. Starfish has the ability to regenerate lost arms, or even regenerate and grow into a new starfish from just one arm.

Rabbit

This refers to the rabbits' population growth problem of Fibonacci. If a pair of rabbits can produce one pair of rabbits (one male and one female) every month, every new born rabbit can produce another pair of rabbits in their third months, and the rabbits never die, what is the total number of rabbits after 50 months?

Chambered Nautilus

Try to draw the cross section of a chambered nautilus in a rectangle that has a ratio of 1:1.618. First, draw a square inside a rectangle. Then in the adjacent smaller rectangle that you can see, draw another square inside it. Keep drawing it in the same way until no more squares can be drawn. Finally, draw a spiraling curve through every square and it will result in a Golden Spiral.

Sunflower

Imagine putting one seed in the centre of the sunflower, and another near the centre. Then put another one near the centre by turning 0.618 of a circle (360°× 0.618 = 222.48° ). Continue putting the seeds with this method to fill up the whole flower core, forming a unique arrangement of the sunflower seeds. In fact, 0.618:1 = 1:1.618 is the Golden Ratio.

Growth Rings of Trees

When chopping down trees, the growth rings on the cross section of the tree trunk are revealed. They represent the age of the tree. The rings in lighter color are formed in spring, whereas those in darker color are formed in autumn and winter. However, trees that grow in tropical regions may not have growth rings due to minimal seasonal changes.