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## The gallery groove

All of the geometry that has been used to form the Sun and the Sun line is currently static and now needs to be transfered onto the dynamic geometry system of the pyramid. Whilst doing this, the original CAD drawing with the incorrectly placed internal architecture will be used because the building's architecture is designed so that the Sun system aligns with the incorrect surveying.

It is helpful to itemise the steps that were taken in the formation of the static Sun point so that they can be exactly replicated when forming the corresponding dynamic geometry:

1. The ascending passage line was drawn in the geometry and the hidden geometry start point identified
2. An angle adjustment was made to the ascending passage line by placing it theoretically into the gallery groove and applying the gallery end wall mathematics to it
3. The adjusted ascending passage line angle was then added to the geometry starting at the hidden geometry start point, to create the gallery bench line (which is why the original CAD drawing needs to be used)
4. The bend point in the hexagonal rod was projected onto the gallery bench line to form the position of the Earth
5. The Sun was then located 200 cubits down the gallery bench line from the Earth point
6. The 1 radian line from the Sun that connects to the main chamber roof was added

These steps now need to be reproduced in the dynamic geometry.

The interactive drawing shows the ascending passage line highlighted which terminates at the hidden geometric start point at the top of the ascending passage.

### 1) Moving the ascending passage line

So far, when the ascending passage line has been positioned in the groove to perform the calculations within the gallery it has been done theoretically. This manipulation now has to be done physically in the geometry and the ascending passage line needs to be transfered into the groove. This task is performed by starting at the only available point, the hidden geometry start point, and then using transformation based on the primitive triangle that was explained on the previous page (page E4). From the start point, the hypotenuse of the triangle is added, with a length of 22.1 cubits, and then a vertical offset of 7.7 cubits is added so that the resulting ascending passage line becomes positioned in the groove. If you adjust the geometry in this manner the interactive drawing shows the result, with the transformation line and the gallery groove line both highlighted. (As will be seen after the system is completed, the length of the vertical adjustment does not have any effect upon the system and it also requires alteration when the surveying of the internal architecture is corrected, and so its length is not relevant.)

The effect of this design system in the architecture is that the gallery groove to the right of the green groove line does not technically exist within the geometry and consequently the architects have destroyed the gallery groove line to the right of this point which can be seen in this 360° video of the gallery if you pause it at 33 seconds in and pan around to look at the west (right) wall groove line.

If you now start the clock you can see that the vertical oscillation created by the dynamic movement of the geometry perfectly matches the height of the gallery and explains why the architecture in the gallery is designed in this manner.

Stop the clock and set the rotation angle of the Earth back to its starting position.

### 2) Adjusting the line angle

The angle of the geometry groove line at this stage is still that of the ascending passage and therefore does not match the gallery's angle and needs to be adjusted to the floor angle of the gallery by applying the mathematics of the gallery end walls. Whilst keeping your eye on the groove line, click here to change the angle of the groove line to that of the gallery floor.

### 3) Projecting onto the groove line

With the groove line now adjusted to the gallery floor angle, the Earth point can be projected onto the groove line by drawing a vertical geometric line from the static upper end of the hexagonal rod and marking off the point on the groove line at the intersection. Create the Earth point on the groove line and then start the clock to show how this dynamic geometry moves over time. The vertical line, because it is starting from the static top end of the straight section of the hexagonal rod, always has the same position horizontally in the geometry, and the top of this vertical line, where it intersects with the moving groove line, represents the position of the Earth.

### 4) Positioning the Sun

The next part of the construction is to position the Sun 200 cubits down the dynamic groove line which can be seen by zooming out to display the lower area of the pyramid. The Sun is situated the same horizontal distance from the top end of the hexagonal rod as it was when it was formed in the static geometry, but is now moving up and down in unison with the movement of the dynamic groove line, as would be expected from the manner in which its position has been created.

The final part of the construction is to add the Sun line to the system, constructed at 1 radian to the vertical from the Sun point.

This unusual looking system is not yet complete because there another line that can logically be added to if you think through the gallery's construction carefully. The adjusted groove line that is displayed on the screen was formed by placing the static ascending passage line into the gallery groove and then applying the gallery wall end mathematics to adjust it to the floor angle. When the gallery was analysed there was a second transformation that was also identified at the north end of the gallery where the roof angle was created by running the mathematics from the floor angle at the north end of the gallery to form the checking angle of the roof, which was arcCos(200/221). So this roof checking angle can also be drawn onto the system.

This is performed in exactly the same manner in which the groove floor line was added to the system, by starting the roof checking line from the hidden start point of the groove. If you add the roof checking line you can see resulting construction, where the roof checking line continues down the gallery and eventually intersects with the Sun line where a second Sun drawing has been drawn.

If you look at this second Sun drawing with the system in motion you can see that the Sun is moving up and down the 'vertical' shaft of the moved trial passages. The detail is quite remarkable and if you zoom in to the trial passages area you can see the architect's system clearly. The Sun moves up and down in a not quite vertical manner, and the intersection points of the 1 radian Sun line with the drawing of the Sun perfectly draw out the 'vertical' shaft's walls.

The whole of this system is a complex checking device designed to ensure that the geometry that is being used is a perfect match to the original architect's design, that all of the points in the geometry have been correctly identified, that the surveying has been corrected adjusted and that the gallery end wall mathematics has been correctly applied to the system.

Only when you have got to this stage can the final part of the system be constructed and verified as being correct.

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Last edited: 3rd July 2019
Last code/graphics edit: 29th March 2021