Ohh ..that formula of the number of images between two mirror is for if the mirrors has a hinge (they are connected). So, if there is a hinge and the angle between them is 0 then there is no distance between them, then there will be no image. If they're parallel at a distance then that formula wouldn't work at all, because it isn't ment for that. The number of images is infinite just because of infinite reflections. This case doesn't contradict that, 360/0 is undefined. Simply, that formula is not applicable here.
You are absolutely right! This formula only applies to hinges. It's good you pointed out this. I discussed it here in the case of theta=0. In this situation, it doesn't matter whether you connect them with the hinges or not because even after separating them with a distance, the reflection property remains as it is. So, I am really glad you pointed this out, but here it's just an extension. The formula is not meant for it but there is no flaw in extending it like this at 0°. If you go according to this, then at this condition the object should be thin like a plane object (perfect 2D object) which in real life is not possible. So either you have to build a perfect 2D object or you can remove the hinge (only for this condition) and separate the mirror according to the dimension of the object, maintaining the same property as before. The only thing we have to consider here for this extension is that the hinge is at infinity and the mirror has infinite aperture.
No, they don't. Parallel lines stay at a fixed distance apart forever in Euclidean Geometry. And it's the reality of course. You are correct here and I appreciate it 🙂 But with the hinge and an object between them is more like a projective geometry not a Euclidean one. If you consider it like a projective geometry because the configuration of mirrors is like that and also we cannot have a perfect 2D object in reality, then it becomes more easy to grasp and the hinges would appear to be at infinity. However, in reality infinity doesn't mean the same. It's all about the scale... Sun's rays are not parallel but we consider it for our small purposes. Why? It's for the same reason...
@@vakrahara Projective Geometry?? .. can you give the source or proof where it links projective geometry to that formula? It'll help me a lot. And yes Sun rays are not parallel but, since it's far far away, the rays make an angle which tends to zero, between them. In this context also, if θ tends to 0, (360/|θ|)-1 gives us infinity, which is also an accurate result. But, this is not a perfect parallel case. So, how can we extend this formula to the perfect parallel case when θ=0. Then, we need to dig into the derivation of the formula, so that we can know where it is applicable and where it is not. But, if you give that projective geometry part with proof, then I'll have no more confusion 👍
@@photostudy8035 you are sticked to the formula itself and I am talking about extending it. It's correct that we cannot build such mirrors with infinite apertures. Let me ask you something, what other mathematical model or explanation do you have to support this observation? Euclidean geometry is not applicable here because of the configuration of the mirrors and also because of the object between them. Technically, theta=0 (complete zero) is not possible here. And this is why you are saying that this formula is not meant for that. And you are absolutely right 👍🏻 I have no objections in it. I too consider this. But, I imagined what if we can extend this formula to this situation? Does it hold good or not? The only way to extend it here is by taking the help of projective geometry. There might exist other ways but I am not aware of it as of now. I came up with this on my own. If you know anything to explain this I too want to have that explanation. This projective geometry is cyclical in nature and forms a closed loop but only when you reach infinity which is weird in itself to think about. 😂 I saw, the number of images goes on increasing as it approaches zero. So, I thought of extending it by using this geometry and it matches my current research. I might be completely wrong, I am not too rigid about my claims but isn't it begging to complete the mathematical structure? We say approaches to zero but afraid to say "equals to 0" even after the mathematics is begging to complete him. My attempt is to provide a mathematical model to give 1/0 a meaning which it currently lags. Even to 0/0, 0°, and many other contradictory forms. This approach was initiated by Aryabhatta, Brahmagupta and Bhaskaracharya and I took it too. I might fail in future but I don't want to quit without giving it a try. 🙂 If I succeed, it will help others and if I fail, I will at least gain some more experience in it.
👍👍👍👍
This intuition is really awesome
Thank you ❤️
Ohh ..that formula of the number of images between two mirror is for if the mirrors has a hinge (they are connected). So, if there is a hinge and the angle between them is 0 then there is no distance between them, then there will be no image. If they're parallel at a distance then that formula wouldn't work at all, because it isn't ment for that. The number of images is infinite just because of infinite reflections. This case doesn't contradict that, 360/0 is undefined. Simply, that formula is not applicable here.
You are absolutely right! This formula only applies to hinges. It's good you pointed out this. I discussed it here in the case of theta=0. In this situation, it doesn't matter whether you connect them with the hinges or not because even after separating them with a distance, the reflection property remains as it is. So, I am really glad you pointed this out, but here it's just an extension. The formula is not meant for it but there is no flaw in extending it like this at 0°. If you go according to this, then at this condition the object should be thin like a plane object (perfect 2D object) which in real life is not possible. So either you have to build a perfect 2D object or you can remove the hinge (only for this condition) and separate the mirror according to the dimension of the object, maintaining the same property as before. The only thing we have to consider here for this extension is that the hinge is at infinity and the mirror has infinite aperture.
@vakrahara So, parallel lines meet at infinity in ordinary Euclidean space?? 🙄.. I mean you are talking about reality, that is R³.
No, they don't. Parallel lines stay at a fixed distance apart forever in Euclidean Geometry. And it's the reality of course. You are correct here and I appreciate it 🙂
But with the hinge and an object between them is more like a projective geometry not a Euclidean one. If you consider it like a projective geometry because the configuration of mirrors is like that and also we cannot have a perfect 2D object in reality, then it becomes more easy to grasp and the hinges would appear to be at infinity.
However, in reality infinity doesn't mean the same. It's all about the scale... Sun's rays are not parallel but we consider it for our small purposes. Why? It's for the same reason...
@@vakrahara Projective Geometry?? .. can you give the source or proof where it links projective geometry to that formula? It'll help me a lot.
And yes Sun rays are not parallel but, since it's far far away, the rays make an angle which tends to zero, between them. In this context also, if θ tends to 0, (360/|θ|)-1 gives us infinity, which is also an accurate result. But, this is not a perfect parallel case. So, how can we extend this formula to the perfect parallel case when θ=0. Then, we need to dig into the derivation of the formula, so that we can know where it is applicable and where it is not. But, if you give that projective geometry part with proof, then I'll have no more confusion 👍
@@photostudy8035 you are sticked to the formula itself and I am talking about extending it. It's correct that we cannot build such mirrors with infinite apertures.
Let me ask you something, what other mathematical model or explanation do you have to support this observation?
Euclidean geometry is not applicable here because of the configuration of the mirrors and also because of the object between them.
Technically, theta=0 (complete zero) is not possible here.
And this is why you are saying that this formula is not meant for that.
And you are absolutely right 👍🏻
I have no objections in it. I too consider this. But, I imagined what if we can extend this formula to this situation?
Does it hold good or not?
The only way to extend it here is by taking the help of projective geometry. There might exist other ways but I am not aware of it as of now. I came up with this on my own. If you know anything to explain this I too want to have that explanation. This projective geometry is cyclical in nature and forms a closed loop but only when you reach infinity which is weird in itself to think about. 😂
I saw, the number of images goes on increasing as it approaches zero. So, I thought of extending it by using this geometry and it matches my current research. I might be completely wrong, I am not too rigid about my claims but isn't it begging to complete the mathematical structure? We say approaches to zero but afraid to say "equals to 0" even after the mathematics is begging to complete him. My attempt is to provide a mathematical model to give 1/0 a meaning which it currently lags. Even to 0/0, 0°, and many other contradictory forms. This approach was initiated by Aryabhatta, Brahmagupta and Bhaskaracharya and I took it too. I might fail in future but I don't want to quit without giving it a try. 🙂
If I succeed, it will help others and if I fail, I will at least gain some more experience in it.