Discovering Pythagoras' Theorem: As an Infant

Yucong Duan

International Standardization Committee of Networked DIKWP for Artificial Intelligence Evaluation(DIKWP-SC)

World Artificial Consciousness CIC(WAC)

World Conference on Artificial Consciousness(WCAC)

(Email: duanyucong@hotmail.com)

Introduction

From the earliest days of my life, I was fascinated by shapes, patterns, and the spatial relationships between objects. As I played with blocks, observed the environment, and interacted with my surroundings, I began to notice consistent patterns and relationships in the world around me. Driven by curiosity, I sought to understand these geometric relationships more deeply. Through observation, experimentation, and logical reasoning, I embarked on a journey that would lead me to discover Pythagoras' Theorem, a fundamental principle in geometry relating the sides of a right-angled triangle.

In this narrative, I will detail how, starting from basic experiences as an infant, I independently observed, experimented, and logically deduced Pythagoras' Theorem. Each concept evolved explicitly from my experiences, ensuring that my understanding is grounded in reality and free from subjective definitions.

Chapter 1: Exploring Shapes and Patterns1.1 Early Interactions with ShapesPlaying with Blocks

• Observation: I had blocks of different shapes—squares, rectangles, triangles, and circles.

• Experiment: Stacking and arranging the blocks in various configurations.

• Reflection: Each shape had unique properties and fit together in specific ways.

Recognizing Triangles

• Observation: Triangular blocks had three sides and could form interesting structures.

• Semantics: A triangle is a shape with three straight sides and three angles.

1.2 Understanding Angles and CornersNoticing Right Angles

• Observation: Some corners formed a "square" angle, like the corners of my square blocks.

• Experiment: Placing two sticks together to form different angles.

• Reflection: A right angle is a specific type of angle that looks like the corner of a square.

Semantics:

• Right Angle: An angle of 90 degrees, forming a perfect corner.

Chapter 2: Constructing Right-Angled Triangles2.1 Creating Triangles with Right AnglesUsing Objects

• Experiment: Arranging sticks or blocks to form triangles with one right angle.

• Observation: These triangles had one corner that matched the right angle of my square blocks.

Labeling the Sides

• Semantics:

• Hypotenuse: The side opposite the right angle, the longest side.

• Adjacent Sides: The two sides that form the right angle.

2.2 Measuring LengthsUsing Units

• Experiment: Measuring sides using uniform units (e.g., block lengths).

• Observation: I could assign numerical lengths to each side of the triangle.

Chapter 3: Investigating the Relationships Between Sides3.1 Observing Patterns in Right-Angled TrianglesRecording Measurements

• Experiment: Creating multiple right-angled triangles with different side lengths.

• Data Collection:

• Triangle 1: Sides of lengths 3 units, 4 units, and 5 units.

• Triangle 2: Sides of lengths 6 units, 8 units, and 10 units.

• Triangle 3: Sides of lengths 5 units, 12 units, and 13 units.

Noticing Patterns

• Observation: The longest side length seemed related to the other two sides.

• Reflection: There might be a consistent relationship between the side lengths.

3.2 Squaring the Side LengthsCalculating Squares

• Operation: For each side length, calculate the square (multiply the length by itself).

• Example:

• 32=93^2 = 932=9

• 42=164^2 = 1642=16

• 52=255^2 = 2552=25

• Triangle 1:

Observing the Sum of Squares

• Observation: The sum of the squares of the two shorter sides equals the square of the longest side.

• $9+16=25$

• Reflection: This pattern holds true for the other triangles as well.

Chapter 4: Formulating Pythagoras' Theorem4.1 Hypothesizing the RelationshipGeneralizing the Observation

• Hypothesis:

• In any right-angled triangle, the square of the hypotenuse equals the sum of the squares of the other two sides.

Mathematical Expression:

• Notation:

• Let $a$ and $b$ be the lengths of the sides adjacent to the right angle.

• Let $c$ be the length of the hypotenuse.

• Equation:a2+b2=c2a^2 + b^2 = c^2a2+b2=c2

4.2 Testing the HypothesisUsing Different Triangles

• Triangle 2:

• $a=6$, $b=8$, $c=10$

• 62+82=36+64=1006^2 + 8^2 = 36 + 64 = 10062+82=36+64=100

• 102=10010^2 = 100102=100

• Conclusion: $36+64=100$, so a2+b2=c2a^2 + b^2 = c^2a2+b2=c2

• Triangle 3:

• $a=5$, $b=12$, $c=13$

• 52+122=25+144=1695^2 + 12^2 = 25 + 144 = 16952+122=25+144=169

• 132=16913^2 = 169132=169

• Conclusion: $25+144=169$, so a2+b2=c2a^2 + b^2 = c^2a2+b2=c2

Confirmation:

• The hypothesis holds true for all tested right-angled triangles.

Chapter 5: Visual Proof Using Squares5.1 Constructing Squares on Triangle SidesDrawing Squares

• Experiment: Drawing squares on each side of a right-angled triangle.

• Observation: The area of each square corresponds to the side length squared.

Calculating Areas

• Example for Triangle 1:

• Square on side $a=3$: Area =32=9= 3^2 = 9=32=9 units²

• Square on side $b=4$: Area =42=16= 4^2 = 16=42=16 units²

• Square on side $c=5$: Area =52=25= 5^2 = 25=52=25 units²

5.2 Demonstrating the Relationship VisuallyCombining Areas

• Observation: The combined area of the smaller squares equals the area of the largest square.

• $9$ units² $+$ $16$ units² $=$ $25$ units²

• Reflection: This visual representation reinforces the mathematical relationship.

Chapter 6: Extending the Theorem to Other Triangles6.1 Non-Right-Angled TrianglesTesting the Relationship

• Experiment: Measuring sides of triangles without a right angle.

• Observation: The equation a2+b2=c2a^2 + b^2 = c^2a2+b2=c2 does not hold.

• Conclusion: The theorem applies specifically to right-angled triangles.

6.2 Importance of the Right AngleUnderstanding Necessity

• Reflection: The right angle is essential for the relationship to be valid.

• Semantics: Right-angled triangles have unique properties not shared by other triangles.

Chapter 7: Practical Applications of Pythagoras' Theorem7.1 Determining DistancesMeasuring Diagonals

• Scenario: Finding the distance between two points in a square or rectangle.

• Application: Using the theorem to calculate the length of the diagonal.

• Example:

• A rectangle with sides $a=6$ units and $b=8$ units.

• Diagonal length $c$:c=a2+b2=62+82=36+64=100=10 unitsc = \sqrt{a^2 + b^2} = \sqrt{6^2 + 8^2} = \sqrt{36 + 64} = \sqrt{100} = 10 \text{ units}c=a2+b2​=62+82​=36+64​=100​=10 units

7.2 Real-World ProblemsConstruction and Design

• Application: Ensuring structures are built with right angles and accurate measurements.

• Example: Verifying that a corner is square by checking that the sides satisfy a2+b2=c2a^2 + b^2 = c^2a2+b2=c2.

Chapter 8: Reflecting on the Discovery8.1 The Universality of the TheoremConsistency Across Scales

• Observation: The theorem holds true regardless of the size of the triangle.

• Reflection: This suggests a fundamental geometric principle.

8.2 The Power of Logical ReasoningFrom Observation to Proof

• Process: Starting with simple measurements and patterns, I formulated a general rule.

• Semantics: Logical reasoning allows for the discovery of universal truths.

Chapter 9: Exploring Further Implications9.1 Connection to Coordinate GeometryCalculating Distance Between Points

• Concept: The theorem can be used to find the distance between two points on a plane.

• Formula:Distance=(x2−x1)2+(y2−y1)2\text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}Distance=(x2​−x1​)2+(y2​−y1​)2​

• Semantics: This is a direct application of Pythagoras' Theorem in coordinate space.

9.2 Introduction to TrigonometryRelating Sides and Angles

• Concept: Ratios of the sides of right-angled triangles lead to definitions of trigonometric functions (sine, cosine, tangent).

• Semantics: Pythagoras' Theorem is foundational for trigonometry.

Conclusion

Through observation, experimentation, and logical reasoning, I was able to discover and formulate Pythagoras' Theorem. Starting from basic experiences with shapes and measurements, I recognized patterns in right-angled triangles and deduced a fundamental relationship between their sides. By grounding each concept in reality and evolving the semantics explicitly, I arrived at a theorem that not only explains geometric properties but also has wide-ranging applications in mathematics and the real world.

This exploration demonstrates that complex mathematical concepts can emerge naturally from simple observations. By avoiding subjective definitions and relying on direct experiences, profound ideas become accessible and meaningful. Pythagoras' Theorem not only provides insight into geometry but also serves as a cornerstone for further mathematical studies, including trigonometry and coordinate geometry.

Epilogue: Implications for Learning and AI

This narrative illustrates how foundational mathematical principles can be understood through direct interaction with the environment and logical reasoning. In the context of artificial intelligence and cognitive development, it emphasizes the importance of experiential learning and the evolution of semantics from core experiences.

By enabling AI systems to observe patterns, formulate hypotheses, and derive laws from observations, we can foster the development of intuitive understanding similar to human learning. This approach avoids reliance on predefined definitions and promotes the natural discovery of mathematical relationships.

Note: This detailed narrative presents the conceptualization of Pythagoras' Theorem as if I, an infant, independently observed and reasoned it out. Each concept is derived from basic experiences, emphasizing the natural progression from simple observations of shapes and measurements to the understanding of a fundamental geometric theorem. This approach demonstrates that with curiosity and logical thinking, foundational knowledge about mathematics can be accessed and understood without relying on subjective definitions.

References

1. International Standardization Committee of Networked DIKWP for Artificial Intelligence Evaluation (DIKWP-SC),World Association of Artificial Consciousness(WAC),World Conference on Artificial Consciousness(WCAC). Standardization of DIKWP Semantic Mathematics of International Test and Evaluation Standards for Artificial Intelligence based on Networked Data-Information-Knowledge-Wisdom-Purpose (DIKWP ) Model. October 2024 DOI: 10.13140/RG.2.2.26233.89445 .  https://www.researchgate.net/publication/384637381_Standardization_of_DIKWP_Semantic_Mathematics_of_International_Test_and_Evaluation_Standards_for_Artificial_Intelligence_based_on_Networked_Data-Information-Knowledge-Wisdom-Purpose_DIKWP_Model

2. Duan, Y. (2023). The Paradox of Mathematics in AI Semantics. Proposed by Prof. Yucong Duan:" As Prof. Yucong Duan proposed the Paradox of Mathematics as that current mathematics will not reach the goal of supporting real AI development since it goes with the routine of based on abstraction of real semantics but want to reach the reality of semantics. ".