# Thread: Imaging fourth dimension

1. I have studied some things about four dimensional geometry from the internet. I understand what the fourth dimension is and I can draw a hypercube an so on. The problem is that I can't imagine four dimensonal things. Can someone? How can I learn for example to turn a hypercube around in my mind?

If you think you can fully imagine four dimensional things, think again. Can you for example tell how many octahedrons does it take to make a polychron (which means a four dimensional "polyhedron"). You can easily imagine how 4 triangles make a tetrahedron and 8 triangles make an octahedron. What about any fourdimensional object?

2.

3. I have studied some things about four dimensional geometry from the internet. I understand what the fourth dimension is and I can draw a hypercube an so on. The problem is that I can't imagine four dimensonal things. Can someone? How can I learn for example to turn a hypercube around in my mind?
impossible. you cant imagen things perfectly if evolution hasent encountered that physical phenomena.

4. Originally Posted by Zelos
I have studied some things about four dimensional geometry from the internet. I understand what the fourth dimension is and I can draw a hypercube an so on. The problem is that I can't imagine four dimensonal things. Can someone? How can I learn for example to turn a hypercube around in my mind?
impossible. you cant imagen things perfectly if evolution hasent encountered that physical phenomena.
Are you sure? I can imagine many other things I haven't seen in real world.

5. its becuase it isnt completly out of reach for the normal mind. While imaging a galaxy, a soler system, a molecule, a atom in its true form is rather hard, and mostly impossible. 4D is impossible to imagen

6. Originally Posted by Zelos
its becuase it isnt completly out of reach for the normal mind. While imaging a galaxy, a soler system, a molecule, a atom in its true form is rather hard, and mostly impossible. 4D is impossible to imagen
Yeah thats obviously what you THINK but how do you KNOW. I remember hearing some guy saying he could imagine 4D. Where did you find the information or did you think it just by yourself?

7. He is lieing. Human mind evolved in a 3D world and it cant imagen more than that. a flying horse is in our grasp. we know animals flying and a horse. just put them toghater.

4D cant we imagen. End of story so dont even try. you can create a program showing 3D and another dimension as either colors, time or something else that makes it possible to grasp but you dont see its true form then

8. I've heard that people can "smell colors" or "feel voices" or see new colors while using drugs. I just don't get it. What is so supernatural about four dimensions? We can develop arificial intelligence which haven't experienced anything in real life and still can deal with some things quite fine.

9. smell colors and feel voices has to do with the brain mixing senses toghater. so the sight signals for some reason goes off to smell center of the brain.
What is so supernatural about four dimensions
nothing. we just can percept it except by our aging. ask a 2D man to understand 3D and he wont

10. Originally Posted by Massi^-
I've heard that people can "smell colors" or "feel voices" or see new colors while using drugs. I just don't get it. What is so supernatural about four dimensions? We can develop arificial intelligence which haven't experienced anything in real life and still can deal with some things quite fine.
I think you'll find that's a 2-D experience, ie Too Dim.

Didn't the phrase '4D object' originate in the Doctor Who series way back in the 1960's?.

As for 'feeling a voice' is that like standing on somebody's windpipe?

11. its older than that. people dealed with 4D a long time ago. but it was rejected to be truie since you cant get a angle of 90Â° from the oither 3D without bieng one of them

12. It may be true that it's hard to imagine something that's at a right angle to reality, but it's not impossible. I can fairly clearly imagine simple 4D shapes as a continuous series of 3D slices. I can even imagine very simple shapes in up to 6 dimensions. 6D objects can be seen as "pick a 3D point to get a 3D object" or, equivalently, a 3D space within each 3D point. The lack of practical experience with 4+ dimensions says nothing about the limits of the human imagination. It's harder to understand the true scale of the universe than it is to imagine the true shape of a hypercube or hyperball.

13. Zelos is right... we can't imagine something which we haven't experienced. Our ideas come from experiences(and we haven't experienced 4D), they don't originate Ex Nihilo.

I think this would go the same with this statement: "or see new colors while using drugs."

14. It may be true that it's hard to imagine something that's at a right angle to reality, but it's not impossible. I can fairly clearly imagine simple 4D shapes as a continuous series of 3D slices.
then you dont imagen 4D bu 3D. its like the 3D picture of a 4D cube. that we can see and we know its a 4D cube bvut we cant see hjow it is in 4D

The lack of practical experience with 4+ dimensions says nothing about the limits of the human imagination
it does. We evovled in 3D. so our mind never evolved the capability to imagen/percept 4D since it is no survival value in it. THerefor we CANT

It's harder to understand the true scale of the universe than it is to imagine the true shape of a hypercube or hyperball.
This is wrong. they are both hard but theuniverse is possible while a hypercube is impossible.

15. What are you idiots prattling on about...

There is NO such thing as a 4-D cube!
You cannot just add a dimension and keep the name - it's STUPID!

1-D = Line
2-D = Plane
3-D = Cuboid
4-D = Quadic! (I propose)

Now I thought everybody knew that if you could see a Quadic your 3-d eyes (which actually only see in 2-D!) would perceive a perfect sphere with a uniformly flat sinlge surface existing in another dimension!

Problem solved!

16. Originally Posted by MagiMaster
It may be true that it's hard to imagine something that's at a right angle to reality, but it's not impossible. I can fairly clearly imagine simple 4D shapes as a continuous series of 3D slices. I can even imagine very simple shapes in up to 6 dimensions. 6D objects can be seen as "pick a 3D point to get a 3D object" or, equivalently, a 3D space within each 3D point. The lack of practical experience with 4+ dimensions says nothing about the limits of the human imagination. It's harder to understand the true scale of the universe than it is to imagine the true shape of a hypercube or hyperball.
Can you really imagine 4D or is it only like Zelos said "then you dont imagen 4D bu 3D. its like the 3D picture of a 4D cube. that we can see and we know its a 4D cube bvut we cant see hjow it is in 4D " ?

17. yes. what you'll see will be similar to this cube on a flat piece of paper
*__
/_ /|
|_|/

sorry for this lousy drawing

18. What are you idiots prattling on about...

There is NO such thing as a 4-D cube!
You cannot just add a dimension and keep the name - it's STUPID!
dont be stupdier than you are. when you say 4D cube you mean the 4D version of a cube. like a 3D square is a cube

Can you really imagine 4D or is it only like Zelos said "then you dont imagen 4D bu 3D. its like the 3D picture of a 4D cube. that we can see and we know its a 4D cube bvut we cant see hjow it is in 4D " ?
yes i am right.

19. Originally Posted by Zelos
What are you idiots prattling on about...

There is NO such thing as a 4-D cube!
You cannot just add a dimension and keep the name - it's STUPID!
dont be stupdier than you are. when you say 4D cube you mean the 4D version of a cube. like a 3D square is a cube

Can you really imagine 4D or is it only like Zelos said "then you dont imagen 4D bu 3D. its like the 3D picture of a 4D cube. that we can see and we know its a 4D cube bvut we cant see hjow it is in 4D " ?
yes i am right.
Which episode of startrek did you regurgitate to come with a '4d cube'
was it the same one that talked about the 17 sided equalateral two dimensional triangle?

20. Just to settle this little semantic tiff, the family of n-dimensional cubes is called the family of hypercubes - and the "4d cube" mentioned here is called the 4-cube, while the standard cube is the 3-cube. The general n-dimenisional cube for a specific dimension n is guess what, a n-cube.

Any questions?

21. We are the borg
You will be assimilated
We are perfect
We are allways right
Resistence is futile

22. Originally Posted by river_rat
Just to settle this little semantic tiff, the family of n-dimensional cubes is called the family of hypercubes - and the "4d cube" mentioned here is called the 4-cube, while the standard cube is the 3-cube. The general n-dimenisional cube for a specific dimension n is guess what, a n-cube.

Any questions?
I'm really sorry to have to break this to you but mass exists in three dimensions - 4D or whatever else you saw in the Cinema does not exist. I know this a very difficult concept for you juniors to grasp, but in the fullness of time you will see it. - What you might want to consider is multi-dimensional time. Also multidimensional Gravity, Magnetism. And as for photons somebody is gonna break it to you one day that they are two dimensional when they exist outside an Atom and 3-d when within (ie merging with an electron).

23. billco. we arent talking about the extra dimension existing or not in a subatomic level or like that. we are talking about imaging it

24. Originally Posted by billco
I'm really sorry to have to break this to you but mass exists in three dimensions - 4D or whatever else you saw in the Cinema does not exist. I know this a very difficult concept for you juniors to grasp, but in the fullness of time you will see it. - What you might want to consider is multi-dimensional time. Also multidimensional Gravity, Magnetism. And as for photons somebody is gonna break it to you one day that they are two dimensional when they exist outside an Atom and 3-d when within (ie merging with an electron).
and what does any of this have to do with my post?

25. Originally Posted by Zelos
billco. we arent talking about the extra dimension existing or not in a subatomic level or like that. we are talking about imaging it
Do you mean Imaging (as in taking pictures) or Imagining (thinking about it?)

Aren't I allowed to join in? will nobody answer my question about the single dimensional corkscrew? - One day you b'stards will miss me!
Then you'll all say "If only we'd answered his questions, he'd still be here!" 8)

26. rather wondering "did he die or just did he forgot his password as he got more senile" just kididng

27. Originally Posted by Zelos
rather wondering "did he die or just did he forgot his password as he got more senile" just kididng

We are the borg
You will be assimilated
We are perfect
We are allways right
Resistence is futile
Least I got to 70 before I lost my what was it???

28. Originally Posted by billco
Now I thought everybody knew that if you could see a Quadic your 3-d eyes (which actually only see in 2-D!) would perceive a perfect sphere with a uniformly flat sinlge surface existing in another dimension!
If I understood you, what you described would be the 4D equivalent of a cylinder. (One of two possible extensions.)

A 4-cube would look like the same cube at each slice in range (until you started rotating it).

I can actually imagine a 4-cube. I can't imagine an image of a 4-cube since images are 2D or semi-3D, but I have a firm idea of what one is, and how it's shaped. (And there are many things our minds can do that has no survival value. Namely anything called art.)

29. What I really want to know is if I take a single-dimensional figure (a line) and wind it into a corkscrew - how many real dimensions will it have?
Also if I take a 2-D plane and bend it how many dimensions will it have?
Since Time can be 2-d but if you distort it you produce gravity is that then 3-D time or is gravity just the 3rd dimension of Time?

30. Originally Posted by billco
What I really want to know is if I take a single-dimensional figure (a line) and wind it into a corkscrew - how many real dimensions will it have?
Also if I take a 2-D plane and bend it how many dimensions will it have?
1 and 2 respectively, at least they are still dimension 1 and 2 manifolds.

"4-D', 'hypercube', 'hypersphere' and so on are not from science fiction, they're standard in mathematics. Topologists (and others) frequently work with n-dimensional things. For example, the Poincare conjecture has n-dimensional analogues that had all been proven true before the 3-dimensional version.

31. Also, it's possible to twist a line segment through any number of dimensions; however, it's still fundamentally 1-D. The same goes for a plane or a cube (2-D and 3-D, obviously). Even the Klein Bottle is fundamentally 2 dimensional even though it cannot be visuallized in less than 4 dimensions.

32. I like 0 dimensional spaces personally (its where my masters disertation lives) - perhaps we should go through the idea of dimension here for anyone who is interested

33. Here:
http://www.dogfeathers.com/java/hyprcube.html

and here:
http://www.geom.uiuc.edu/docs/outreach/4-cube/

That's what I concluded when I was trying to create a 3d animation of a solid rotating hypercube. When I was baked.
A hypercube grid consists of 8 cubes. It becomes a hypercube, when the walls of the cubes are brought together. Warping the 7 cubes in ones mind is easy up to the last one. To imagine all of them connected, one has to imagine himself inside the last cube, turned inside-out. Then the "outer" side of the hypercube are in fact the inner side of the walls of the 8th one, that is, you have to imagine yourself inside.

34. Was that what you were looking for?

35. http://torina.fe.uni-lj.si/~zlobec/cube/Cube.html
This should also help you visualize it.

36. Is it just me or are we looking at 2D projections of 3D and 4D objects?

My screen is flat. How about yours? :wink:

I think I once read some obscure instructions on how to vaguely imagine a 4D object, maybe by Martin Gardner? Not sure.

Anyway, I like to think as time as the fourth dimension. So any moving 3D object is technically a 4D phenomenon. But that's a cheap answer.

Our eyes deal with 2D very well, because that's all we see: 2D projections on our retina. To make an impression of the third dimension, our brain analyses single and double images (each eye from a slightly different perspective) for an impression of depth. That's complicated enough. There is no mechanism to "see" a fourth spatial dimension, and now you might argue that human brains are really so limited that they cannot imagine something that's radically different from anything they have ever processed.

37. As well as using two eyes to perceive 3-d we also use the focusing mechanism to aid depth judgement, and moving your head (with only one eye open) together with past experience (size of objects etc) also helps, it's very complex.

38. I always thought the 4th dimension was the translation of the 3rd dimension through time? what does adding a few more lines to a 3-d cube do to make it 4-d, like in the exapmles people showed.

39. Originally Posted by Draculogenes
I always thought the 4th dimension was the translation of the 3rd dimension through time? what does adding a few more lines to a 3-d cube do to make it 4-d, like in the exapmles people showed.

glad you said that, thats what i was thinking x-y-z and time, surely the 4D cube is just a more complex 3D object, that still has x-y-z co-ordinates

40. Yeah, 4-D space [I think] is like the emporers clothes, the old "We can see it, you must be stupid if you can't" - I also think there is a mob of philosophers out there who have bets as to just how much bollocks they can get people to accept.

All these guys are really doing is taking the 3 axis and making them 4. On that basis you can make it infinite.

Here's a newer thought for you.

Space is a SINGLE dimensional 3 AXIS quantity.
Time is a SINGLE dimensional 3 AXIS quantity.

Not my theory, can't remember who's, hell of a lot more sense than most though!

41. Originally Posted by captaincaveman
glad you said that, thats what i was thinking x-y-z and time, surely the 4D cube is just a more complex 3D object, that still has x-y-z co-ordinates
A 4d-cube is not a 3d object. It can't be embedded into 3-space.

Do people not learn about n-dimensional euclidean space? I find that hard to believe...anyone who has worked with a system of equations with more than 3 real variables was working in something with more dimensions than R^3.

42. Originally Posted by shmoe
Originally Posted by captaincaveman
glad you said that, thats what i was thinking x-y-z and time, surely the 4D cube is just a more complex 3D object, that still has x-y-z co-ordinates
A 4d-cube is not a 3d object. It can't be embedded into 3-space.

Do people not learn about n-dimensional euclidean space? I find that hard to believe...anyone who has worked with a system of equations with more than 3 real variables was working in something with more dimensions than R^3.
I don't usually Quote Wiki, but since I know some of you swear by it.....

Originally Posted by http://en.wikipedia.org/wiki/Euclidean_geometry

It also is no longer taken for granted that Euclidean geometry describes physical space.

43. A 4d-cube is not a 3d object. It can't be embedded into 3-space.
... but it can be "projected" into 3D-space. I remember watching a 3D animation (I mean real 3D, i.e. stereoscopic), of a projection of a rotating hypercube. The simplest projection along the fourth axis just removes the fourth coordinate.

44. Projected yes - embedded no.

As you saw with the projection of the hypercube into R^3 - it was a self intersecting mess. The same thing happens with the klein bottle, you cant embedd it into R^3 (even though it is only 2 dimensional!)

45. Originally Posted by billco
I don't usually Quote Wiki, but since I know some of you swear by it.....

Originally Posted by http://en.wikipedia.org/wiki/Euclidean_geometry

It also is no longer taken for granted that Euclidean geometry describes physical space.
So what's your point? Is this somehow relevant to what I said? I don't recall talking about anything at all in "physical space".

46. Originally Posted by M
A 4d-cube is not a 3d object. It can't be embedded into 3-space.
... but it can be "projected" into 3D-space. I remember watching a 3D animation (I mean real 3D, i.e. stereoscopic), of a projection of a rotating hypercube. The simplest projection along the fourth axis just removes the fourth coordinate.
You can likewise project a 3-cube into 2-space.

Another option is to look at 3-d cross sections of the 4-cube. Just like you can look at 2-d cross sections of a 3-cube. Go read "Flatland", there was a description of what a flatlander would see as a 3-sphere passed trough flatland (not difficult to imagine though, but it was an amusing book).

47. Hmm i was reading up on the fourth dimension and i found this. It's quite interesting. Visualizing in 10 dimensions? interesting concepts.

http://www.tenthdimension.com/flash2.php

48. Originally Posted by captaincaveman
Originally Posted by Draculogenes
I always thought the 4th dimension was the translation of the 3rd dimension through time? what does adding a few more lines to a 3-d cube do to make it 4-d, like in the exapmles people showed.
glad you said that, thats what i was thinking x-y-z and time, surely the 4D cube is just a more complex 3D object, that still has x-y-z co-ordinates
In fact, 4D object do have x, y and z coordinates. They also have w coordinates. A 4D hypercube has 16 vertices. The pictures with the fourth set of lines don't actually show a 4D hypercube. They show a 4D hypercube projected on to a 2D surface. Therefore, the lines don't really point in the right directions. What you have to do is imagine the line first stretching up, then left, then away, then another direction that you can't physically see or feel. (If you can understand the concept, you can imagine it thought.)

49. river_rat: As you saw with the projection of the hypercube into R^3 - it was a self intersecting mess.
Actually... if I remember right, what looks like an intersecting mess when projected on a 2D screen looked much more appealing as a stereoscopic movie. I remember images of a weirdly convoluting tube of some sort, kind of like a sock continuously turning inside out. Does anyone have a link to such a movie, maybe one of those red/green ones to be viewed with 3D glasses?

50. Never seen that version - will keep an eye open for it.

51. Originally Posted by MagiMaster
Originally Posted by captaincaveman
Originally Posted by Draculogenes
I always thought the 4th dimension was the translation of the 3rd dimension through time? what does adding a few more lines to a 3-d cube do to make it 4-d, like in the exapmles people showed.
glad you said that, thats what i was thinking x-y-z and time, surely the 4D cube is just a more complex 3D object, that still has x-y-z co-ordinates
In fact, 4D object do have x, y and z coordinates. They also have w coordinates. A 4D hypercube has 16 vertices. The pictures with the fourth set of lines don't actually show a 4D hypercube. They show a 4D hypercube projected on to a 2D surface. Therefore, the lines don't really point in the right directions. What you have to do is imagine the line first stretching up, then left, then away, then another direction that you can't physically see or feel. (If you can understand the concept, you can imagine it thought.)
yeah, im struggling to get my head round it at moment :-D

52. If the brain is working at full capacity and is able to see something more than when it was operating at average capacity. Would that be considered a higher dimension or hallucination?

53. Originally Posted by starlight
If the brain is working at full capacity and is able to see something more than when it was operating at average capacity. Would that be considered a higher dimension or hallucination?
I have this theory that when you are blind drunk, and walk into a wall or something that isn't there, that's when you discover the fourth spatial dimension.

54. It would not be possible to imagine 4D instantly. We only have eyes to see two point of view on any object, eventhough we are awared that there are other parts of the object that we cannot see.

To be able to see 4D, we have to be able to see all around an object instantly which means we need more Eyes and we have to surround the objects. But our imagination breaks down as we try to approach the fourth dimension.

Some day when evolution gives us two more eyes on the back of our head, and add another dimension to our brain, then we might be able to imagine or see 4D.

That's my opinion.

55. You might find this link interesting:
http://www.superliminal.com/cube/cube.htm

You can download and install a 4D cube puzzle and on the page you can find links to a 5D cube and even a 2D puzzle!

56. Adding what Zelos and others have said..

You cannot imagine 4D images. Think of a 2D Man being placed in a 3D world. He would not see what we saw, he would just see 2D images in our 3D world. Instead of a human figure, it would be a combinations of mostly circles and some squares. Same with an airplane, or anything. If a 4D object entered our world, or we entered the 4D dimension, it would just be a blob to us, we could not see what it is.

57. I don't think we can imagine four dimensional objects. But maybe we can create the illusion, such as in M.C.Escher's ascending and descending.

http://britton.disted.camosun.bc.ca/...descending.jpg

58. Originally Posted by Cat1981(England)
I don't think we can imagine four dimensional objects. But maybe we can create the illusion, such as in M.C.Escher's ascending and descending.

http://britton.disted.camosun.bc.ca/...descending.jpg
dont need to think. it is ismpossible a bieng can never imagen more dimension than its able to percept wich is never more than the amount of spatial dimensions in the universe

59. Am I wrong here, but is the fourth dimension not time ? Therefore motion ? If so what is wrong with imagining a rubix cube moving around ?

60. I think the topic is about a 4th spatial dimension. Time doesn't qualify.

61. Am I wrong here, but is the fourth dimension not time ? Therefore motion ? If so what is wrong with imagining a rubix cube moving around ?
You are both right and wrong at the same time. There is the fourth temporal dimension which is time (note that describing it just as "time" is wrong) and the fourth spatial dimension also known as the 5th dimension.

62. Im sorry, but this isnt making sense to me now any more. ANOTHER spacial dimension ?

And for the purpouse of ??

63. I always use this analogy when I go to imagine a 4th or 5th dimension.

Letâ€™s pretend we are in a 2d plane (like stick figures drawn on a sheet of paper), and some how we exist in a conscious state. We can easily point in 2 dimensions, i.e. up and down, and left and right.

Now, from the point of view of an observer of this 2d world, i.e. a person looking at the piece of paper, it is easy to see how it is impossible for the stick figure, not only to point in the direction of the third dimension (out or into the page), but even comprehend what it is like.

For us to build some mental image of a 4th or 5th dimension is equally as difficult.

64. This may sound Stupid but I always thought the 4th dimension was time. So you have a 3d shape and its 4th dimension is how that objects changes as time continues. Please if Iâ€™m wrong, which I think I am, can someone explain 4d for me.
Thanks.

65. I always thought that was the case as well. My post refers to any extra spatial dimensions.

66. Originally Posted by Lee W
This may sound Stupid but I always thought the 4th dimension was time. So you have a 3d shape and its 4th dimension is how that objects changes as time continues. Please if Iâ€™m wrong, which I think I am, can someone explain 4d for me.
Thanks.
the theory of Relativity has 3 spacial and 1 time dimension.

thats not to say that every 4 dimensional vector space has a 4th dimension which must be labeled a time dimension, in fact theres not much that mathematicaly separates space and time dimensions, its mainly the physical discription that causes them to differ.

67. Thanks for clearing that up.

68. Also, in GR you are working with locally minkowskian manifolds and not straight forward vector spaces (a HUGE difference)

69. Originally Posted by wallaby
the theory of Relativity has 3 spacial and 1 time dimension.
So has "real life", surely? Time is an inescapable dimension, and we find 3 spatial dimensions sufficient for most purposes. Or have I misunderstood the discussion here?

70. Originally Posted by river_rat
Also, in GR you are working with locally minkowskian manifolds and not straight forward vector spaces (a HUGE difference)
Showing off, river_rat? Why not explain what you mean?

71. Originally Posted by Guitarist
Originally Posted by wallaby
the theory of Relativity has 3 spacial and 1 time dimension.
So has "real life", surely? Time is an inescapable dimension, and we find 3 spatial dimensions sufficient for most purposes. Or have I misunderstood the discussion here?
that was my point, relativity is an application of some mathematical ideas to this universe, thats an application to one object. in a more theoretical view time need not always be the 4th dimension does it?

Originally Posted by river_rat
Also, in GR you are working with locally minkowskian manifolds and not straight forward vector spaces (a HUGE difference)
so many more years of learning to go.

72. Originally Posted by Guitarist
Originally Posted by river_rat
Also, in GR you are working with locally minkowskian manifolds and not straight forward vector spaces (a HUGE difference)
Showing off, river_rat? Why not explain what you mean?
Ah, it's much more fun to make profound statements then it is to explain them Can i assume that everyone knows what a manifold is or should i start a thread on geometry and topology?

73. Originally Posted by river_rat
Can i assume that everyone knows what a manifold is
I doubt it.
or should i start a thread on geometry and topology?
Yea, that would be fun. But recall I tried a "topology primer" thread, which (apart from you biting my head off a coupla times) evoked little interest.

But I should enjoy such a thread (you could always link back to my primer for the intro stuff, if you think it's kosher)

74. Sounds like a plan - ill try dig up your old thread and resurrect it

75. Here, imagine ten dimensions.

76. I find this discussion a little funny:P

It's strange to try to imagine 4+ dimentions, because you don't even know if it exists.

I think this question depends on if mathematics is something the humans have invented or ifit's something we have discovered. If it's something we have invented, then I dont think that there is any 4th dimention, but if its something we have discovered, and discover more and more, then it is possible.

77. I think you have mis-read the title, it is imaging not imagining - a subtle difference, since we can represent 3 dimensions on a 2-D screen why not 4?

78. I got it wrong after reading all of those pages too. LOL!

79. Time for bed....

80.

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