Offering a couple of geometric paradoxes, who told me a couple of classic paradox questions?

I don't know what kind of geometric paradox you want, whether it's for math or art.

You can find a lot of them.

1 "Any triangle is isosceles" (see Fig. 1).

Let abc be an arbitrary triangle, make the bisector of c and the perpendicular bisector of ab, and let the intersection of the two lines be e. From E to AC and BC perpendicular EF and EG, and even EA and EB.

Now, the right triangles CFE and CGE are congruent, and each right triangle has CE as the common hypotenuse, and FCE GCE (defined by the angle bisector), CF CG.

At the same time, the right-angled triangles EFA and EG are congruent, the right-angled side FE of one triangle is equal to the right-angled side EG of the other (the bisector of c is equidistant from both sides of the angle), and the hypotenuse of one triangle EA is equal to the hypotenuse EB of the other triangle (any point E on the perpendicular bisector of the line segment AB is equidistant from the two endpoints of the line segment). ∴fa＝gb

From the above two points:

CF FA CG GB (Equal Amount Equal Amount).

i.e. ca cb

That is, this triangle is isosceles.

2 "A right angle is equal to an obtuse angle" (see Figure 2).

Let ABCD be an arbitrary rectangle, and make a line segment of the same length as BC be, outside the rectangle, so that it is also equal to AD.

Make the perpendicular bisector of de and ab: they are perpendicular to the non-parallel lines, and they must intersect at a point p. Connect AP, BP, DP, EP.

At an equal distance from any point on the perpendicular bisector of a line segment to the two endpoints of the line segment, pa pb, pd pe. In addition, according to the graph, ad be, in apd and bpe, the three edges correspond to each other, so apd and bpe are congruent. DAP EBP, however, BAP is the base angle of the isosceles triangle APB, BAP ABP.

dap bap ebp abp (equal amount).

i.e. DAG EBA

That is, a right angle is equal to an obtuse angle.

The geometric paradox constructs a pattern that makes a figure that only exists in a two-dimensional plane world, and is an image that cannot exist in a three-dimensional three-dimensional world through three-dimensional painting techniques such as sketching and line drawing.

The "Impossible Steps" was invented by British geneticist Leoniel S. Penrose and his son, mathematician Roger Penrose, who published it in 1958 and is often referred to as the "Penrose Steps".

In this step, the highest and lowest steps can never be found, and there is never an end to the "impossible steps" ...

Zeno's paradox. Half the time equals twice the time.

ABCD has the same velocity and the same magnitude, 5678 moves to the right, ABCD moves to the left, 8 to 4 is equal to the time A reaches 1, but 5678 moves in half the time of ABCD (because only two squares are moved relative to 4), and the time of C series movement is double that of B series (because it moves four squares relative to 5678).

Just look up Zeno's paradox on the Internet!It's classic.

The barber's paradox, Russell's paradox.

There was a barber in a certain city, and his advertising slogan read: "I am very skilled in barbering, and I am famous all over the city." I will shave the faces of all those in the city who do not shave their faces, and I will only shave their faces.

I extend a warm welcome to all of you!"There is an endless stream of people who come to him to shave their faces, and naturally they are all those who don't shave their faces. One day, however, the barber saw in the mirror that his beard had grown, and he instinctively grabbed the razor

If he doesn't shave himself, he belongs to the "one who doesn't shave himself", he has to shave himself, and what if he shaves himself?He belongs to the "person who shaves his face", so he should not shave his face.

A paradox is when there are two opposing conclusions and outcomes implied in the same proposition or reasoning on the surface, and both conclusions can be justified.

It describes a farmer who is worried that his award-winning cow has gone missing. At this time, the milkman arrived at the farm, and he told the farmer not to worry because he saw the cow in a nearby clearing.

A paradox is defined as the apparent implicit presence of two opposing conclusions in the same proposition or reasoning, both of which are self-justifying.

A paradox is when there are two opposing conclusions implicit in the same proposition or reasoning, and both of them can be justified. The abstract formula for the paradox is that if event A occurs, then it is deduced that it is not A, and if it does not occur in A, it is deduced that A.

Paradox is the confusion of different levels of thinking, meaning (content) and mode of expression (form), subjectivity and objectivity, subject and object, stupidity and value implied in propositions or reasoning, the asymmetry between the content of thinking and the form of thinking, the subject of thinking and the object of thinking, the level of thinking and the object of thinking, and the asymmetry of the structure of thinking and the structure of logic.

All paradoxes arise from the formal logic way of thinking, and the formal logic way of thinking cannot discover, explain, or solve logical errors. The so-called paradox solution is to use the symmetrical logical thinking mode to discover and correct the logical errors in the paradox.

Using symmetrical logic to solve the paradox of God's creation of stones:

Solution: Can God make a stone that he can't lift himself, the energy here is literally the same as God's omnipotent energy, but the connotation is different, it is not the same concept, and it violates the law of uniformity of formal logic. Of course, the stone made by God Almighty can be lifted by itself, and the energy here is the energy of objective ability, which belongs to the category of objectivity;

Can God make a stone that he can't lift on his own? Whether or not here means whether or not can belong to the category of subjective wishes. Therefore, this so-called paradox is the result of confusing two concepts that are literally the same and have different connotations, without looking at the propositional language form and not looking at the propositional thinking content.

As long as the two concepts are separated, this paradox can be solved.

Formalizing traditional logic will inevitably lead to paradoxes that only look at the language form of propositional questions and ignore the content of propositional thinking; Only symmetrical logic can distinguish the thinking form of a concept from the thinking content, so as to solve the contradiction. From the point of view of symmetrical logic, this paradox is purely a linguistic game – a violation of the law of identity in formal logic.

A paradox is a statement or proposition that seems logical on the surface, but in fact it is self-contradictory, untenable, and a mistake in thinking. Paradoxes are challenges to Hyomin's logical thinking, and are often used in thinking and research in fields such as philosophy, mathematics, physics, linguistics, etc. The existence of paradoxes reveals the limitations of language and logic, highlights the limitations and inadequacies of our thinking and reasoning, and reminds us to pay attention to examining various premises and starting points when thinking about problems and reasoning.

Paradoxes often show people's logical dilemmas and ideological biases, and some wrong ways of thinking can be broken through paradoxes, so that we can understand the essence of things more clearly.

Paradoxes come in many forms, the most famous of which are the following:

1.Bertran's Paradox: If a coin is uniformly random, what is the probability that at least half of the 100 tosses will be heads?

Bertrand's paradox is that when you think the probability is 50%, the probability is actually about. This is due to the fact that in the case of a uniform distribution, when the more you toss, the higher the probability, so a 50% probability is a misconception.

2.Achilles and the Tortoise Paradox: Achilles must overtake a tortoise, but must start in a queue 10 meters in front of him.

First, Achilles had to reach the end of the queue where the turtle was, which required him to run 10 meters, and by this time the turtle had advanced 1 meter and he was now 9 meters behind Achilles. Next, Achilles had to run twice as fast in order to catch up with the tortoise, but then the tortoise would move forward again, and Achilles was still a metre away from the tortoise. No matter how fast Achilles ran after that, when he reached the tortoise again, the tortoise would always move a certain distance, and Achilles would have to start running from that distance.

Coincidentally, this paradox is logically contradictory.

3.Infinite Soul Paradox: This paradox says that if a person is in a room hollowed out and there is a closed container in it, and there is a tiny ball in this container, so if the person is a reasoner, then he will think that the ball cannot be removed.

Because the ball is placed in a completely closed container, and the position of the ball cannot be observed by a person, it is likely that the ball is no longer in the container. But if the person opens the container, he can spot the ball. Therefore, this paradox is also self-contradictory.

The existence of paradoxes makes people aware of the complexity of reasoning and thinking, and at the same time motivates people to delve deeper into logic and cognitive science to better understand the nature and application of paradoxes. In our daily lives, we must also pay attention to these paradoxes in order to avoid contradictory opinions or wrong judgments, and to promote the development of correct ways of thinking and reasoning.