# Solution to 4D Geo

Austin Lei, Evan Chen, Moor Xu, Nathan Wong

This puzzle consists of four different types of “geo”, which comprise the four main steps of the puzzle:

1. Geometry problems, as suggested by the title;
2. Geography, subtly suggested by “where” in the flavortext;
3. Geology, subtly suggested by “hard” in the flavortext;
4. Geomancy, subtly suggested by “even” and “odd” in the flavortext and from the chart given at the end of the puzzle.

We present these four steps in order below.

## 1. Geometry

The first step is to solve all the triangle geometry problems, which requires some computation in Cartesian coordinates.

We comment briefly on the mathematical problem of recovering the third vertex of a triangle given two vertices and the center. In what follows, triangle ABC has A and B given. Then:

• Given the centroid G, the vertex G is recovered by C = 3G − (A+B).
• Given the orthocenter H, the vertex C is exactly the orthocenter of triangle HAB.
• Given the incenter I, the vertex C is recovered by reflecting the line AB across lines AI and BI and taking the intersection.
• The hardest sub-problem is to find the vertex C given the symmedian point K. In fact, such a vertex C is not unique in general — there can be up to two — which is why the problem specifies that the triangles are all acute. Fortunately, the problem can be found on Google. See Constructing the Triangle from Two Vertices and the Symmedian Point by Michel Bataille, in Forum Geometricorum Volume 18 (2018) 129-133. The completed table is shown here.

#P.xP.yQ.xQ.yR.xR.yS.xS.y
#1 (G)-32.071-60.641171.316-92.3666.258-264.47948.501-139.162
#2 (I)-243.537179.039338.676341.390-30.976-36.228-31.320122.610
#3 (H)27.908-102.106439.054296.796-73.114203.114-17.421145.711
#4 (I)-243.537179.03911.549104.939-30.976-36.228-56.79267.804
#5 (H)27.908-102.106439.054296.79612.826-85.74618.153-91.237
#6 (K)79.667101.833-20.136-72.144-17.585114.72422.19088.200
#7 (K)79.667101.83333.673-114.227-17.585114.72433.94289.051
#8 (H)41.500-82.870439.054296.796-73.114203.1143.950122.419
#9 (I)-243.537179.039338.676341.39042.320-120.81920.520106.335
#10 (K)44.091-70.301-20.136-72.144-17.585114.7249.098-61.302
#11 (G)15.056-80.64445.409-83.7906.258-264.47922.241-142.971
#12 (I)47.190-91.380338.676341.390-30.976-36.22835.932-30.136
#13 (K)79.667101.833-20.136-72.14448.400-90.35019.049-70.431
#14 (H)27.908-102.10649.478-94.461-73.114203.11434.156-99.532
#15 (G)15.056-80.644171.316-92.3666.258-264.47964.210-145.830
#16 (G)15.056-80.644171.316-92.36667.56933.39084.647-46.540

## 2. Geography

Each of the (X,Y) pairs represents a latitude and longitude pair. We map out which city/lake/farm/etc. is indicated by each pair.

#LatitudeLongitudeLocation name
#1-32.071-60.641Diamante
#2-31.320122.610Talc Lake
#3-17.421145.711Topaz
#411.549104.939(Koh Pich) Diamond Island
#512.826-85.746Apatite
#622.19088.200Diamond Harbour
#733.673-114.227Quartzsite
#841.500-82.870Gypsum
#942.320-120.819Quartz Mountain
#1044.091-70.301Mount Apatite
#1145.409-83.790Calcite
#1247.190-91.380Corundum Point
#1348.400-90.350Topaz Lake
#1449.478-94.461Quartz Island
#1564.210-145.830Quartz Lake
#1667.56933.390Apatity

Note that the X-values are simply in increasing order. This suggests we will likely want to re-order these later.

## 3. Geology

Each of the locations corresponds to the name of a mineral on the Mohs scale of mineral hardness. We record the value of the Mohs entry here.

#CenterMissingMineralMohs
#1GPDIAMOND10
#2ISTALC1
#3HSTOPAZ8
#4IQDIAMOND10
#5HRAPATITE5
#6KSDIAMOND10
#7KQQUARTZ7
#8HPGYPSUM2
#9IRQUARTZ7
#10KPAPATITE5
#11GQCALCITE3
#12IPCORUNDUM9
#13KRTOPAZ8
#14HQQUARTZ7
#15GSQUARTZ7
#16GRAPATITE5

At this point we can begin to fill out the shield chart: plugging in the i’th hardness into mi and evaluate the arithmetic.

RowFigure positionCalculationIndex
Row 1Fourth daughter8 - 5 =3
Row 1Third daughter2 + 9 =11
Row 1Second daughter1 =1
Row 1First daughter5 =5
Row 1Fourth mother (K)7 - 3 =4
Row 1Third mother (I)9 - 3 =6
Row 1Second mother (H)7 - 1 =6
Row 1First mother (G)8 + 5 =13
Row 2Fourth niece10 - 7 =3
Row 2Third niece10 - 3 =7
Row 2Second niece5 + 7 =12
Row 2First niece8 - 7 =1
Row 3Left witness10 - 9 =1
Row 3Right witness9 =9
Row 4Judge7 - 1 =6

This gives the following in the shield chart.

 3 11 1 5 4 6 6 13 3 7 12 1 1 9 6

## 4. Geomancy

Sort the table of triangle centers and Mohs hardnesses by the triangle center and the omitted point. The letters G, H, I, K are the "usual" letters in Euclidean geometry for the centroid, orthocenter, incenter, and symmedian point, respectively.

#CenterMissingMineralMohsParity
#10KPAPATITE5odd
#7KQQUARTZ7odd
#13KRTOPAZ8even
#6KSDIAMOND10even
#12IPCORUNDUM9odd
#4IQDIAMOND10even
#9IRQUARTZ7odd
#2ISTALC1odd
#8HPGYPSUM2even
#14HQQUARTZ7odd
#5HRAPATITE5odd
#3HSTOPAZ8even
#1GPDIAMOND10even
#11GQCALCITE3odd
#16GRAPATITE5odd
#15GSQUARTZ7odd

Placed these into the four houses in the upper right of the geomancy shield; following geomancy tradition we only care about the parity of the hardnesses.

For example: the head, neck, body and feet of the first mother (in the cell labeled G) correspond to the parities of the Mohs ratings for the centroid G, for P, Q, R, S respectively. As 10, 3, 5, 7 are even, odd, odd, odd, this gives Caput Draconis.

After the four figures (called mothers) are placed, executing the geomancy procedure will produce the four daughters, and then similarly the rest of the chart. The completed chart is shown below, with the names of the geomantic figures annotated as well.

 1212Amissio 1112Cauda Draconis 1121Puer 2211Fortuna Major 1122Fortuna Minor 1211Puella 2112Conjunctio 2111Caput Draconis 2122Rubeus 1112Cauda Draconis 2111Caput Draconis 2221Tristitia 1212Amissio 2112Conjunctio 1122Fortuna Minor

Finally, take the numbers from the third step and index them into the name of the figures in the present step. Then read the chart from left to right, top to bottom:

RowFigure positionIndexParitiesSymbol nameLetter
Row 1Fourth daughter31212AmissioI
Row 1Third daughter111112Cauda DraconisN
Row 1Second daughter11121PuerP
Row 1First daughter52211Fortuna MajorU
Row 1Fourth mother (K)41122Fortuna MinorT
Row 1Third mother (I)61211PuellaA
Row 1Second mother (H)62112ConjunctioN
Row 1First mother (G)132111Caput DraconisS
Row 2Fourth niece32122RubeusB
Row 2Third niece71112Cauda DraconisR
Row 2Second niece122111Caput DraconisI
Row 2First niece12221TrisitiaT
Row 3Left witness11212AmissioA
Row 3Right witness92112ConjunctioI
Row 4Judge61122Fortuna MinorN