Honeybees communicate the location of food sources through an intricate waggle dance that relies on mathematics. The dance uses the positions of the sun and gravity to indicate direction and distance in a way that allows other bees to efficiently find the food. Hexagonal tiling is the most efficient shape for honeycomb cell construction because hexagons use the least amount of material while providing strength and stability. Mathematical formulas like Euler's help explain relationships in the hexagonal structures bees create.

2. INTRODUCTION
Today we are going to give a presentation on the theme mathematics around us
You must seen that mathematics is everywhere around us but have you observed
Mathematics in nature. Yes nature also brings mathematics along with it. The gift of
Math's is not only give humans but there is one more species which opts this talent
Yes and they are honeybees you might think that how honeybees are connected
With math's but believe me you will be shocked to hear their mathematical mindset.

3. This discovery was first made by Karl von Frisch before him no one have ever
Thought the idea that how honeybees communicate .He has done many experiment
To understand the secrets behind their mathematical communication for this he
Kept a plate of sweet water in his garden and when a honeybee of his garden come
To drink he put a mark the bee. In a few hours he observed that the others bees of
Garden are come to drink the sweet water. It was surprisingly event because he know
Well that honeybees doesn't follow each other rather than each and every bees is
Well aware about the exact location of the sweet.

4. WAGGLE DANCE
This is a diagram called waggle dance this is a diagram which represent the
Dance sun plays a very crucial role in this system. The light coming from the
Sun gets polarized I the atmosphere and bees having a good sense ability
Detects the exact location of the sun even looking at the opposite direction
Just like a compass. They use the angle of sun to indicate general direction of
Food ‘s location they also use gravity as reference why they dance straight
Upwards it means fly towards the sun and when they move downward it
Mean fly away from it similarly left and right convey the exact coordinates in
Space which is km away. The duration of dance fraction of seconds indicates
The length of time to reach there for like 1sec=1km . Calculate the wind speed
-to more finally calibrate the position. Thus can be done at any time of the year
So bees can do the math.

5. HEXAGONAL TILING
Hexagonal tiling, also known as hexagonal tessellation, is a
geometric arrangement of regular hexagons that completely
covers a two-dimensional plane without leaving any gaps or
overlaps. In other words, it's the process of tiling a surface
using hexagonal tiles in such a way that they fit together
seamlessly. This tiling technique is commonly observed in
various natural and man-made structures, with honeycombs
being one of the most well-known

6. WHY DO HEXAGONS?
Honeycombs are composed of hexagonal cells primarily because hexagons are an efficient
Shape for filling space with minimal material while providing it stability. Hexagons are
Most efficient shape because of several reasons.
1. Equal sides and angles
2. Efficient use of space
3. Even distribution of stress
4. Strength and stability

7. THE HONEYCOMB CONJECTURE
The Honeycomb Conjecture, also known as the Honeycomb Problem or
Honeycomb Conjecture, is a mathematical puzzle and problem in geometry that
pertains to the efficient way in which bees construct honeycombs using
hexagonal cells This conjecture implies that, when given a certain amount of
material or resources to create partitioning cells, the hexagonal arrangement will
require the least amount of material for the partition In 1999, mathematician
Thomas Hales, with the assistance of computers, successfully proved
the Honeycomb Conjecture. Hales' proof involved intricate mathematical
reasoning and optimization techniques.

8. EULER ‘S FORMULA
Euler's formula is a fundamental relationship in mathematics that relates the number of vertices,
edges, and faces of a polyhedron (a three-dimensional solid with flat faces) to a constant value.
This formula has deep connections to graph theory, topology, and geometry. Euler's formula is
typically stated as: V-E+F=2
Where:
• V represents the number of vertices (corners) of the polyhedron.
• E represents the number of edges (straight-line segments connecting vertices) of the
polyhedron.
• F represents the number of faces (flat surfaces) of the polyhedron.
Euler's formula is most commonly associated with polyhedral, but it can also apply to other
types of structures, such as planar graphs. It provides a powerful tool for analyzing and
understanding the relationships between the various components of these structures.

9. TESSELLATIONS AND SYMMETRY
Tessellations: Tessellations, also known as tiling's or mosaics, are arrangements
of shapes or tiles that completely cover a surface, such as a plane, without
any gaps or overlaps. These shapes can be regular polygons (like squares,
triangles, and hexagons) or irregular shapes. The key idea is that they fit
together in a way that creates a repeating pattern. Tessellations have been
used in art, design, and architecture for centuries and are an intriguing topic
in mathematics.
Symmetry: Symmetry is a concept that describes a balanced or harmonious
arrangement of elements in a pattern or object. It signifies a sense of order
and proportion. Symmetrical objects or patterns have certain operations or
transformations that leave them unchanged.