The Fourth Dimension Mary Jaskowak Mercyhurst University May 2017 Mary Jaskowak (Mercyhurst University) The Fourth Dimension May 2017 1 / 16
How do you move in one dimension? -5-4 -3-2 -1 0 1 2 3 4 5
How do you move in one dimension? +5-5 -4-3 -2-1 0 1 2 3 4 5
How do you move in one dimension? +5-5 -4-3 -2-1 0 1 2 3 4 5-3 Mary Jaskowak (Mercyhurst University) The Fourth Dimension May 2017 2 / 16
What about in the second dimension? 3 2 1-5 -4-3 -2-1 1 2 3 4 5-1 -2-3
What about in the second dimension? 3 2 1-5 -4-3 -2-1 1 2 3 4 5-1 -2-3
What about in the second dimension? 3 2 1-5 -4-3 -2-1 1 2 3 4 5 +3-1 -2-3
What about in the second dimension? 3 2 1 +2-5 -4-3 -2-1 1 2 3 4 5 +3-1 -2-3
What about in the second dimension? 3 2 3 + 2i 1 +2-5 -4-3 -2-1 1 2 3 4 5 +3-1 -2-3
What about in the second dimension? 3 2 3 + 2i 1 +2-5 -4-3 -2-1 1 2 3 4 5 +3-1 -2-3
What about in the second dimension? 3 2 3 + 2i 1 +2-5 -4-3 -2-1 1 2 3 4 5 +3-1 -2-3
What about in the second dimension? 3 2 1 3 + 2i 2 + i +2-5 -4-3 -2-1 1 2 3 4 5 +3-1 -2-3
What about in the second dimension? 3 2 1 3 + 2i 2 + i +2-5 -4-3 -2-1 1 2 +3 3 4 5-1 3 + 2i + 2 + i = 5 + 3i -2-3 Mary Jaskowak (Mercyhurst University) The Fourth Dimension May 2017 3 / 16
Rotations in the second dimension -4-3 -2-1 1 2 3 4
Rotations in the second dimension 3 + 4i -4-3 -2-1 1 2 3 4
Rotations in the second dimension (3 + 4i) x i = 4 + 3i 3 + 4i -4-3 -2-1 1 2 3 4
Rotations in the second dimension (3 + 4i) x i = 4 + 3i 3 + 4i -4-3 -2-1 1 2 3 4
Rotations in the second dimension (3 + 4i) x i = 4 + 3i -4 + 3i 3 + 4i -4-3 -2-1 1 2 3 4 Mary Jaskowak (Mercyhurst University) The Fourth Dimension May 2017 4 / 16
So how do we rotate in the third dimension? Mary Jaskowak (Mercyhurst University) The Fourth Dimension May 2017 5 / 16
So how do we rotate in the third dimension? William Hamilton (1843) Mary Jaskowak (Mercyhurst University) The Fourth Dimension May 2017 5 / 16
So how do we rotate in the third dimension? William Hamilton (1843) Mary Jaskowak (Mercyhurst University) The Fourth Dimension May 2017 5 / 16
Had he only known... In 1877, Frobenius proved that the only finite dimensional, associative division algebras have dimension 1, 2, or 4 R, C, H If we do not require associativity, we gain one last finite dimensional division algebra, the octonians, which have dimension 8. Mary Jaskowak (Mercyhurst University) The Fourth Dimension May 2017 6 / 16
The Quaternions The quaternions have the form: a + bi + cj + dk Mary Jaskowak (Mercyhurst University) The Fourth Dimension May 2017 7 / 16
The Quaternions The quaternions have the form: with: a, b, c, d ɛ R a + bi + cj + dk Mary Jaskowak (Mercyhurst University) The Fourth Dimension May 2017 7 / 16
The Quaternions The quaternions have the form: a + bi + cj + dk with: a, b, c, d ɛ R i 2 = j 2 = k 2 = ijk = 1 Mary Jaskowak (Mercyhurst University) The Fourth Dimension May 2017 7 / 16
The Quaternions The quaternions have the form: a + bi + cj + dk with: a, b, c, d ɛ R i 2 = j 2 = k 2 = ijk = 1 Non commutative, associative algebra Vector space, 4 dimensional over R Mary Jaskowak (Mercyhurst University) The Fourth Dimension May 2017 7 / 16
The Quaternions The quaternions have the form: a + bi + cj + dk with: a, b, c, d ɛ R i 2 = j 2 = k 2 = ijk = 1 Non commutative, associative algebra Vector space, 4 dimensional over R By multiplying these 4-dimensional numbers we can rotate things in the third dimension. Mary Jaskowak (Mercyhurst University) The Fourth Dimension May 2017 7 / 16
The Quaternions As a 4 4 matrix, the quaternion q = a + bi + cj + dk can be written as a b c d A = b a d c c d a b d c b a The conjugate of the quaternion q would be A T. The quaternion q = a + bi + cj + dk can also be represented as a 3 3 matrix: a 2 + b 2 c 2 d 2 2bc 2ad 2bd + 2ac 2bc + 2ad a 2 b 2 + c 2 d 2 2cd 2ab 2bd 2ac 2cd + 2ab a 2 b 2 c 2 + d 2 With this representation, quaternionic multiplication is compatible with the multiplication of 3 3 real orthogonal matrices (MM T = I ) Mary Jaskowak (Mercyhurst University) The Fourth Dimension May 2017 8 / 16
Normed Algebra The quaternions are a normed algebra with and q = a 2 + b 2 + c 2 + d 2 q 1 q 2 = q 1 q 2 A quaternion q with q = 1 is called a unit quaternion. Mary Jaskowak (Mercyhurst University) The Fourth Dimension May 2017 9 / 16
Rotations in R 3 Let q = a + bi + cj + dk be a unit quaternion. 1 The axis of rotation is the line through the origin and (b, c, d) 2 The angle of rotations is θ where a = cos(θ/2) Mary Jaskowak (Mercyhurst University) The Fourth Dimension May 2017 10 / 16
Rotations in R 3 Let q = a + bi + cj + dk be a unit quaternion. 1 The axis of rotation is the line through the origin and (b, c, d) 2 The angle of rotations is θ where a = cos(θ/2) The matrix associated to a unit quaternion takes the point (x, y, z) R 3 to (x, y, z ) in the usual way: a 2 + b 2 c 2 d 2 2bc 2ad 2bd + 2ac x x 2bc + 2ad a 2 b 2 + c 2 d 2 2cd 2ab y = y 2bd 2ac 2cd + 2ab a 2 b 2 c 2 + d 2 z z Mary Jaskowak (Mercyhurst University) The Fourth Dimension May 2017 10 / 16
Rotations in R 3 Example: The unit quaternion that represents a rotation of 90 about the x-axis would be q = 1 + 1 1 0 0 i which has matrix Q = 0 0 1 2 2 0 1 0 Mary Jaskowak (Mercyhurst University) The Fourth Dimension May 2017 11 / 16
So what is the fourth dimension? Euclidean space (R 4 ) 4 space dimensions coordinates needed distance 0 iff points are equal distance non-negative Figure: The tesseract is the four-dimensional cube. Mary Jaskowak (Mercyhurst University) The Fourth Dimension May 2017 12 / 16
So what is the fourth dimension? Euclidean space (R 4 ) 4 space dimensions coordinates needed distance 0 iff points are equal distance non-negative Minkowski space: 3 space like dimensions, 1 time like coordinate free distinct points can have distance 0 distance can be negative Figure: Minkowski spacetime is a four-dimensional manifold, referred to as a light cone. Mary Jaskowak (Mercyhurst University) The Fourth Dimension May 2017 12 / 16
Spacetime Figure: Spacetime as a loaf of bread is a way Einstein explained the idea of time as another physical dimension. Mary Jaskowak (Mercyhurst University) The Fourth Dimension May 2017 13 / 16
God as the Fourth Dimension Verse John 20: 19-23 Reference...the doors were locked...jesus came and stood in their midst... Mary Jaskowak (Mercyhurst University) The Fourth Dimension May 2017 14 / 16
God as the Fourth Dimension Verse John 20: 19-23 John 20: 26-29 Reference...the doors were locked...jesus came and stood in their midst......jesus came although the doors were locked, and stood in their midst... Mary Jaskowak (Mercyhurst University) The Fourth Dimension May 2017 14 / 16
God as the Fourth Dimension Verse John 20: 19-23 John 20: 26-29 Acts 8: 39-40 Reference...the doors were locked...jesus came and stood in their midst......jesus came although the doors were locked, and stood in their midst......the Spirit of the Lord snatched Philip away, and the eunuch saw him no more... Mary Jaskowak (Mercyhurst University) The Fourth Dimension May 2017 14 / 16
God as the Fourth Dimension Verse Genesis 5:24 Reference Enoch walked with God, and he was no longer here, for God took him. Mary Jaskowak (Mercyhurst University) The Fourth Dimension May 2017 14 / 16
God as the Fourth Dimension Verse Genesis 5:24 Hebrews 11:5 Reference Enoch walked with God, and he was no longer here, for God took him. By faith Enoch was taken up so that he should not see death, and he was found no more because God had taken him. Mary Jaskowak (Mercyhurst University) The Fourth Dimension May 2017 14 / 16
God as the Fourth Dimension Verse Genesis 5:24 Hebrews 11:5 John 10:39 Reference Enoch walked with God, and he was no longer here, for God took him. By faith Enoch was taken up so that he should not see death, and he was found no more because God had taken him. they tried again to arrest him; but he escaped from their power Mary Jaskowak (Mercyhurst University) The Fourth Dimension May 2017 14 / 16
God as the Fourth Dimension Verse Genesis 5:24 Hebrews 11:5 John 10:39 John 8:59 Reference Enoch walked with God, and he was no longer here, for God took him. By faith Enoch was taken up so that he should not see death, and he was found no more because God had taken him. they tried again to arrest him; but he escaped from their power so they picked up stones to throw at him; but Jesus hid and went out of the temple area. Mary Jaskowak (Mercyhurst University) The Fourth Dimension May 2017 14 / 16
Closing Remarks With all these theories out there we have a good idea of what the fourth dimension might be but we will never be able to fully comprehend what it is. Mary Jaskowak (Mercyhurst University) The Fourth Dimension May 2017 15 / 16
Closing Remarks With all these theories out there we have a good idea of what the fourth dimension might be but we will never be able to fully comprehend what it is. May the Fourth be with you. Mary Jaskowak (Mercyhurst University) The Fourth Dimension May 2017 15 / 16
Questions? Mary Jaskowak (Mercyhurst University) The Fourth Dimension May 2017 16 / 16
Thank you Mary Jaskowak (Mercyhurst University) The Fourth Dimension May 2017 16 / 16