Poetry: The Human Condition

We are all full of hypocrisies.
Small phrases
Little words
Actions
That do not align.

Letting temporary feelings
Lack of understanding
Control the situation

Pent up anger
Bitterness
Grudges
Explode

Hallelujah
You say
Jesus can forgive all sins
Damn you
You say
I see no good in or from you

Proclaiming to not be of this world
Be better
See clearer
Live well
Yet
That's the typical story
Isn't it?

Show me how you're different

You can't

Our morality does not align
Do not argue morals
Our ethics do no align
Do not speak of your ethics
Our values are not the same
Do not tell me what you value
Beliefs mean nothing on their own
Show me why they matter

Stop using old mantra
I've heard it a thousand times
And most of them said it better than you
Understood it clearer than you
Stop acting like your words are something new

Leave uncertainty at the door
I want your experiences
Not your thoughts
Where is the data, evidence, proof?

Citing a story that a million people have read
Interpreted in more ways than there are people
Dictated differently every day
Basing its claims on how you feel
How you think
Using it as justification

No

It should be the other way

Stop saying that your beliefs are logical
They are just your personal thoughts
If you truly meant them,
You'd live them

But you don't

Harassing others for the weakest sin of sex
When the strongest sin of pride
Is firm on your shoulders

Didn't you learn from your Jesus?
Humility is the prize
You are worthless
You are replacable
You are dust
No one deserves life
No one deserves love
No one is better than anyone else
(grovel at the feet of your master)

There is no guarantee of salvation
Unless
Thought and action align
Parallel to the grace and will of Above

This is known at death
Not sooner

To make that choice is a curse
To live knowing
Mother cries eternally for the fallen
Father grieves for those who forsake life
Brother is the one that calls them
The one they turn from
You are safe to watch others make the wrong choice

Stop thinking about you
It's never been about you
It never will

Your life is for others
Created to worship
Created to suffer

What do you think the end is?
All your dreams satisfied?
No
The end is the will of the Maker
Reverence for grace and power
Praise to the one who made the rules
Designed the skies

Don't want it?
Don't take it.

Many would rather choose their paths
Better to suffer your own fate
Than prosper the one given

That's the choice
Let them make it

Don't pound on my door
Don't hand me some pages
Don't offer to waste my time

It's all been tried before

You do it for you
Not for me
You do it for your institution
Not for the creater
You do it for self esteem
Not to be faithful

This is the human condition.

Do not call me apathetic

Essay: Lie Algebras

1 Introduction

Back in the 19th century, a mathematician named Sophus Lie introduced the subject of studying continuous symmetries of geometric objects and their derivatives to the mathematical world. These topics are now called Lie groups and Lie algebras respectively.

Since then, these topics have been far reaching in their capabilities; finding a role in many areas such as geometric objects, matrices, and symmetry to name a few. In addition to this, Lie algebras have also found use in several fields of study in physics such as string theory and quantum mechanics.

However, since the topic is still relatively new and is usually taught isolated from other mathematic classes, the capabilities of this subject are still unknown. In this paper, some of the basic elements and associations of Lie algebras will be discussed to give a brief background on this diverse topic.

2 Basic Properties

A Lie algebra is a vector space over a field that multiplies in such a way that a and b hold true for elements X,Y,Z and where [ ] represents multiplication in the vector space. See [1] pg 3, [4] pg 3, and [6] pg 25.
a. [X, X] = 0
b. [X[Y, Z]]+[Y[Z, X]]+[Z[X, Y]] = 0 which can also be written as:
[X[Y, Z]] + [Y [Z, X]] + [Z[X, Y]] = 0
adX.[Y, Z] = [adX.Y, Z] + [Y, adX.Z]
ad[X, Y]=adXadY − adYadX

These properties can be broken down by name. Suppose L is a Lie algebra and F is a field.
a. Bilinear Product
LxL = L
b. Antisymmetry or Skew Symmetry
[X,Y] = − [Y, X] for X,Y ∈ L
c. Linearity
[λX, Y ] = λ[X, Y ], [X + Y, Z] = [X, Z] + [Y, Z] for X,Y,Z ∈ L and λ ∈ F
d. Jacobi Identity
[X[Y, Z]] + [Y [Z, X]] + [Z[X, Y ]] = 0 for X, Y, Z ∈ L

2.1 Importance
Lie algebras are heavily used in particle physics and in string theory. They are also applied in differential geometry, Lie groups, as well as in Hamilton and Quantum mechanics. While these are the main uses for Lie algebras, they can be applied to most fields of mathematics. [7]

3 Lie Groups

Before Lie algebras can be understood, Lie groups should be explained as Lie algebras originated as a way to study Lie groups. A Lie group is simply a topo- logical group that is a manifold. This means that a Lie group is merely a group that differential calculus may be applied to.

3.1 Groups
To be a group, certain properties must be met. [9], pg 172.
a. ForA,B ∈ G, AB ∈ G.
b. For A,B,C ∈ G, A(BC) = (AB)C
c. There is an identity I such that IA = AI = A for every A ∈ G.
d. For every A in G, there exists B = A−1 such that AA−1 = A−1A = I

To satisfy the requirements to be specifically a Lie group, the operations must also be differentiable. A morphism of Lie groups is a smooth map which follows f(gh) = f(g)f(h) and f(1) = 1

Remark: To be smooth means to have a derivative for every order in the domain.

For example, SL(2, R) =

for k, l, m, n ∈ R and kn − lm = 1
is a simple real Lie group.

Non zero real numbers with multiplication are also Lie groups.
On the other hand, an uncountable direct sum of a finite cyclic group would not be a Lie group.

3.2 Connections to Lie Algebras
A Lie algebra is the tangent space of a Lie group at the unit element. [1], pg 16. Put another way, a Lie algebra is a logarithm of a Lie group and a Lie group is an exponential of a Lie algebra. Since a Lie algebra is linear, it is often easier to manipulate than its Lie group.

Suppose L is a Lie algebra of Lie group, G. Then the structure of G near the identity is determined by L.

Example
Let G be a Lie group and let A, B ∈ gl(n) where gl(n) is the subspace of real nxn matrices.
Then eA+B = lim k → ∞(eA/keB/k)k.
Lie(G) = A ∈ gl(n):etA  ∈ G, ∀t ∈ R.
Let A, B ∈ Lie(G).
Then et(A+B) = lim k → ∞(etA/ketB/k)k ∈ G.
Therefore, A + B ∈ Lie(G).
If A ∈ Lie(G) then αA ∈ Lie(G).
Therefore Lie(G) is a vector subspace of gl(n) and closed under scalar multiplication. [8], pg 4.
QED

4 Jacobi Identity

The Jacobi Identity is a large part of what makes a Lie algebra just that. How- ever, in the 20th century is was discovered that this aspect of a Lie algebra has uses separate from Lie Theory. [3]

The Jacobi has been applied to many things including:
The Double Dualization Functor
Tangent-Vector-Valued Differential Forms
Schwartz Distributions

Each of these subject fields are very complicated and would take up too much space to explain here. Basic Theorems regarding their relation to the Jacobi Idenity can be found in [3].

4.1 Example 1
Prove (A × B) × C + (B × C) × A + (C × A) × B = 0
Keep in mind that the cross product is not associative meaning that A × (B × C) ≠= (A × B) × C

Proof
(A × B) × C + (B × C) × A + (C × A) × B = (AC)B − (BC)A + (BA)C − (CA)B + (CB)A − (AB)C = 0
QED

4.2 Example 2
Let L be an algebra over F.
[XY ] = XY − Y X defines L to be a Lie algebra because the Jacobi Identity holds.

The Jacobi Identity may also be proved using the Poisson bracket. The Poisson bracket is very important in dynamics and when the bracket is not met because the Jacobi Identity is not fulfilled, those systems (coined almost- Poisson) become associated with dissipation and waste.

5 Basic Theorems

To gain a better understanding of the complex relations that Lie algebras may find themselves in, a few examples of basic theorems is shown below. [2]

Theorem 1
Let L be a Lie algebra in field F. Let I be a subideal of L.
a. If I is a nilregular ideal of L then N(I) ⊆ N(L)
b. If I is a nilregular subideal of L and every subideal of L containing I is nil- regular, then N(I) ⊆ N(L)

Corollary
Let L be a Lie algebra over a field F. Then every minimum ideal of L is abelian, simple, or irregular.

Theorem 2
Let L be a Lie algebra.
a. If L is a solvable primitive Lie algebra then all core-free maximal subalgebras are conjugate.

b. If A is a self-centralizing minimal ideal of a solvable Lie algebra, L, then L is primitive, A is complemented in L, and all components are conjugate.

Theorem 3
Let L be a Lie algebra.
L is semisimple if and only if its Killing form in nondegenerate. [4], pg 107

5.1 Lie’s Three Theorems
Lie I
If G is a connected and solvable linear algebraic group defined over an algebraically closed field and p : G → GL(V ) a representation on a nonzero finite-dimensional vector space, V, then there is a one-dimensional linear subspace L of V such that p(G)(L) = L.

Lie II
Let G and H be Lie groups with Lie algebras g = Lie(G) and h = Lie(H) such that G is simply connected. If f : g → h is a morphism of Lie algebras, then there is an unique morphism F : G → H of Lie groups lifting f such that f = Lie(F ).

Lie III
The functor Lie is essentially surjective on objects. For every dimensional real Lie algebra g there is a real Lie group G such that g ∼= Lie(G). Moreover, there exists a G which is simply connected.

Lie III is arguably the most challenging of the three because it requires so much structure theory of Lie algebras to prove and does not apply to general Banach-modeled Lie groups and Lie algebras.
Lie I and Lie II relate to the infinitesimal transformations of a transformation group acting on a smooth manifold. Lie III in contrast, states the Jacobi Iden- tity for the infinitesimal transformations of a local Lie group. [5]

6 Conclusion
Lie algebras are a very complex subject that relate to many fields of study. At first glance, this subject field did not seem that intriguing or deep but with more research and understanding of its applications, Lie algebras end up being quite interesting.

Due to its wide use, it was difficult to give a brief summary on the topics it related to because so many of the theorems and proofs that showcase Lie algebras would need a lot of explanation due to their varied and advanced material. For example, the Weyl group is heavily influenced by Lie algebras yet they were not discussed in this paper due to their complexity. Another topic of note that was not brought up are Free Lie algebras which are quite interesting to learn about.
In short, a person who studies higher level math or physics would be hard pressed to not run into Lie algebras at one point in time or another– even if they did not realize it.

As a side note, writing math papers is a lot more challenging than anticipated and it will be a long while before I complain about one of my textbooks being difficult to read. Also, it is very satisfying to spend hours writing strange code and then for it to turn into beautiful math. The least fun aspect of this project would have to have been reading hundreds of pages on a topic and learning so much about it and feeling like you understand it pretty well but do not really know what to write because none of your peers spent days reading about how Lie algebras affect quantum mechanics. The end.

Bibliography

1. [1]  H. Samelson. Notes on Lie Algebras. (Stanford, California, 1989).
   
2. [2]  D. Towers. Chief Factors of Lie Algebras. (Lancaster University, England, 2016).
   
3. [3]  H. Nishimura. The Jacobi Identity beyond Lie Algebras. (University of Tsukuba, Japan, 2009).
   
4. [4]  J. Bernstein. Lectures on Lie Algebras. (Stockholm University, Sweden 2012).
   
5. [5]  S. Sternberg. Lie Algebras. (Harvard, Massachusetts, 2004).
   
6. [6]  A. Kirillov, Jr. Introduction to Lie Groups and Lie Algebras. (Stony Brook University, New York, 2008).
   
7. [7]  H. Georgi. Lie Algebras in Particle Physics. (Ebook, Westview Press, 1999).
   
8. [8]  A. Alexanderian Matrix Lie groups and their Lie algebras (University of Texas, Texas, 2013).
   
9. [9]  T. Hungerford. Abstract Algebra: An Introduction (Saint Louis University, Missouri, 2014).
   

Essay: PBS Secrets of the Dead: Aztec Massacre

SYNOPSIS

400 skeletons in a mass grave were found at Zultepec— an Aztec site, so Archaeologist Elizabeth Baquedano was brought in to analyze the grave. The Aztecs frequently used their enemies as human sacrifices in their religious practices and the skeletons showcased clear signs of dismemberment and frequent absence of certain bones. Many of the skeletons were found in shallow graves outside the temple where religious ceremonies would have taken place. The bones were dated to the early 1500’s. Analyzing the skulls, around 40 of the skulls did not belong to the local tribes but rather Europeans. Based on artifacts at the gravesite, the Europeans were likely to be Spaniards with horses. A historian was spoken to about the Spanish influence, stating that they were small groups of entrepreneurs. The Aztec prince believed that the Spaniards came to fulfill an ancient prophecy so the Aztecs gave the Spaniards small gifts that the Spanish leader, Cortez, wrote about. In the historical record, the Spaniards annihilated the Aztecs with little resistance since the prince refused to admit that the Spanish were not there to fulfill the prophecy however there were Spanish bodies in the mass grave indicating fallacy in the recorded history. Further analysis showed that ten skulls belonged to European females which indicated that they belonged to the second group of Spanish to arrive, the ones that came to arrest Cortez for leaving without permission so Cortez attacked them, taking several captives before journeying back to the Aztecs where the prince had died. Many local groups joined Cortez on the way back to fight against the Aztecs but slowed travel so he left the group behind and went ahead with a smaller group of soldiers, leaving the large abandoned group defenseless to the Aztecs to be captured, kept for months, and then sacrificed to their gods to keep the world in balance. Sacrifices were performed only by the priests who held down the sacrifice, tore out the heart and offered the heart to the sun. The heads were then punctured on each side so that they could be hung next to each other. Half of the skulls found in the grave marked in this fashion were determined to belong to Europeans. Another historian was approached to see if there is a record of European sacrifices by the Aztecs. A sketch of European men and horse heads on the head racks was found, linking the evidence to history. Once sacrificed, the bodies were dismembered and the long bones were given to warriors once stripped of flesh. Some of the bones were cremated and others chewed on by humans, proving the common theory that Aztecs partook in cannibalism. It took Cortez two years to conquer the Aztecs and help found Mexico City once back in the Spanish King’s favor. This proves that the Aztecs fought against the Spanish invaders and sometimes even won before their ultimate defeat.

ANALYSIS

This film compared to the other one showed Archaeology very dependent upon historical texts and knowledge of the locals to deduce the reality of the bones, seemingly looking for clues and patterns that were believed to exist rather than viewing the bones impartially and coming to conclusions from what they deduce. In the other film, the archaeologists analyzed the bones and formulated hypotheses on their findings and knowledge of the local populations then compared their findings with history to see how they fit together. It would have been nice to have an answer to their posed question of when European diseases impacted the Aztecs and local tribes. Using molecular archaeology could be a tool in this discovery. (Kelly and Thomas page 214) This film was focused on the Europeans found present but there was still information to be learned from the natives that were also sacrificed. Perhaps to some extent the order of the skeletons being scarified and placed in the grave may have also been determined. In Kelly and Thomas, it mentions that finding exotics in a grave could indicate status (page 239) so I also wonder if the bones taken from the sacrifices to honor their warriors would be present in their grave sites. Honestly, this film felt more incomplete than the first one and I wasn’t really sure what they were trying to prove until the end when the film stated that the Aztecs had indeed fought back against the Spanish invaders because evidence of the Spanish being sacrificed and eaten by the Aztecs was found. In the first film, I didn’t really have anything to add once complete but this one just left me with so many questions and it’s hard to say how strong the deductions of the skeletons were because of the major text usage in the situation.

SOURCES

1. Kelly, Robert and Thomas, David. Archaeology. Cengage Learning, 2016.

Essay: PBS The Great Incan Rebellion

SYNOPSIS

On the coast of Peru existed the Incan Empire, innovators of their time who created Machu Picchu and gold masterpieces, while being masters of conquest. When the archaeologists Guillermo Cock dug a test trench, several graves were found in even intervals in a crouched position facing East with offerings as expected of Incan burials but near the surface, graves broke the pattern by facing West and lying as if thrown in and covered in simple garments without offerings. In addition, the unusual graves had indications of extreme violence that had previously had not been seen in any Indian grave site such as piercing wounds, smashed skulls, and many broken bones. These injuries brought forth the hypothesis that these individuals could have been stabbed from horseback by Spanish conquistador. An x-ray was used to determine if one skull had been impacted by a bullet but no metal residue was found so two experts on gunshot wounds were brought in to examine the skull. The  analysis lead the experts to believe that it could have been a gunshot wound so the bone plug and the area surrounding the hole had small fissures analyzed, finding small traces of iron, a metal often used with muskets. Since the Spanish conquistador were illiterate and an official record of events was recorded some time later by scribes, it was speculated that parts of the historical account were left out or exaggerated, minimizing the role of the Indian allies. Examining the skeleton of the one that they determined to be the leader, many of the injuries found line up with the Spanish account including crushing wounds to the torso region and the piercing wounds in the skull. However, the other bodies present do not: often showing blunt force trauma to the skull that would not have come from the Spanish but rather other Indians—only three bodies indicate death by Spanish weaponry. In addition to this, some of the bodies found belonged to women. A historian was brought in who confirmed that the Spanish fought with a large number of Indians as allies and that instead of large battles, most of what occurred was skirmishes between Indians based on other documents and witnesses. The leader of the Spanish conquistador had a strong alliance with an Indian tribe in Peru who offered a young girl to be his wife. When the Incans attacked the conquistador, the girl’s mother provided them with a large army who attacked the Inca. It is likely that these graves show such a skirmish. Due to the ongoing conflicts, the dead could not have been properly buried after battle. Once the Incan conquest was finished, the Spanish forgot their allies.

ANALYSIS

The grave site situation was handled very expertly. The archaeologists who worked together were strong work colleagues who were heads in their field of studying Incan history. In addition to this, several experts were brought in to evaluate aspects that differed from a normal Incan grave site. The two men brought in to examine the skull to see if it was a gunshot wound even stated that their purpose was to try to prove the hypothesis wrong but encountered too much evidence to draw any conclusion but that of a musket shot. The archaeologist examined different weaponry on record for the Indian tribes and the Spanish for that time period and concluded what type of weaponry related to which bodily trauma. In addition to all of this, a historian talked with who had an extensive knowledge of this history and using a variety of resources, provided a more likely story than the one the Spanish provided, correlating with what was present in the graveyard. Several forms of analysis was used on the skeletons including skeletal analysis, x-rays, and symbolism of grave patterns, offerings, and remnants of clothing. In Kelly and Thomas’ book, grave site evaluation is discussed more artificially than the methods outlined in the film and the process that was used to determine the reasoning of the graveyard was thorough.

SOURCES

1. Kelly, Robert and Thomas, David. Archaeology. Cengage Learning, 2016.

Essay: Archaeological Materials

INTRODUCTION
Over the course of two days, four samples of lithic materials, three samples of ceramics, and two ground stones were analyzed. The ceramic samples had all of their items evaluated while the lithic material samples specified to choose a representative variety from what was offered. The ground stone station had many metates, grinding substances, and manos to choose two of each from to evaluate.

PROCEDURE

On day one, Camron and I went to the lithic materials and ceramic stations and took pictures of the artifacts presented in the different bags. He lined up two rulers to be an x-axis and y-axis then arranged the items so that they were all within those boundaries then I took a picture. Due to the condensed sections that the different stations were at and the number of other students at these tables, this took almost the full lab time to do so the ground stone station was left for later. It was noted that most groups that were drawing the artifacts had only managed one station. The measurements outlined on page 136 (Kelly) for lithics are obvious to view in the photos.

On day two, Kacey, Liz, and I picked a ground stone, measured it, and ground wheat berries, commenting on the process. Then, we chose a second ground stone and repeated the process with quick cook barley. With the last 15 minutes of class that remained, Liz and I looked up definitions in the textbook and completed the calculations.

CERAMICS

For set 1, the linear punctate and the two stamped designs present were likely to be made with hand tools. None of the potsherds were shiny or had an indication of a finish so the pottery that these sherds came from was more likely built for daily function rather than decoration or occasional use. It is also probable that several of the potsherds belong to the same vessel. The textured pieces would belong on the body and the plain at the base for functionality and appearance purposes. Since the pieces all relatively have the same slope when lying flat, it is unlikely that any of the pieces belong to the rim where a steeper slope is more likely present.

For set 2, the parallel lines were likely to be done by hand since the distance between lines varies slightly from line to line. Since the remnants of paint are clustered, it is also likely that the sherds were hand painted. Since the breaks for these sherds are similar to each other and the coloring is consistent between them, it is likely that these sherds belong to the same vessel, further indicated by an apparent likeness in material. Due to the cracks in the sherds, the paint, and their indicated interior placement, it is likely that these sherds were part of a vessel that held liquids. The paint would have acted as a barrier to preserve the material and the cracks show its use. These sherds have been indicated to be interior pieces due to their coloration and texture. Based on their slopes and markings, these pieces would be in the body of the ceramic since base pieces would be flat and rim pieces would have received more sun exposure in addition to having more design.

For set 3, the sherds are a variety of sizes, thickness, and shape. The designs on three of the sherds are all distinguished and utilize varying color so it is unlikely that the collection of sherds belong to the same vessel. The edges of these sherds also differ from each other: the cylinder piece is smooth but other sherds have jagged edges, smooth edges, or a combination. Due to the durability of the designs on the sherds, the pottery was fired after it was painted. The cylinder piece may have been part of a handle or lid while the painted pieces were likely part of the body of the vessel. The plain pieces look burnt and are fairly flat so they are expected to be part of a vessel base. Due to the wide variation of the sherds presented, functionality is harder to determine. The area that they were found was likely a place that needed many vessels.

The ceramic samples came from blending sites in Utah. Since we only looked at three samples it is unlikely that the selection represents a regional pattern given the limited quantity and variation amongst the items. Ceramics are often dated with thermoluminescence to determine the amount of time since the materials were heated or exposed to sunlight. Dating ceramics is important because knowing the time when the material was last fired is a strong indicator of when the item was in use. Generally, pottery is clay that has been molded into shape, dried, then fired with a finish. By examining the intentional texture and decoration of the pieces, how the ceramic was molded can be determined. These hypotheses could be tested by attempting to assemble the sherds into a ceramic object, performing chemical analysis on the components, and assessing the fragments within the site universe instead of analyzing them out of context.

 LITHIC MATERIALS
The lithic materials viewed were all from Logan, Utah. Due to the sheer number in the collection sets and the variety expressed in size, shape, and color it is likely that these materials formed  a regional pattern. Lithic materials are often dated using Carbon 14 but other methods can be used depending on the material. (Kelly, page 122) If only a quarter of the lithic materials can be dated, then a representative sample in terms of type, material, and size should be taken and dated. The three quarters not dated can then be matched against the average date and compared to the ones similar that were dated.

In set 1, the artifacts were projectile points of large size of varying material. In set 2, the artifacts were projectile points of small size of varying material. In set 3, the artifacts were large bifaces of varying material. In set 4, the artifacts were scrapers of varying material. Attributes are characteristics that distinguish artifacts from each other (Kelly, page 358) but criteria are the standards by which something is measured. For example, the criteria for set 1 is to be a projectile point of a certain size but the attributes would be the material, color, jaggedness of edges, etc. In this way, it is important that artifacts should be considered in terms of type (criteria) first then materials (attributes).

GROUND STONE

A metate is a large, flat stone that is used to grind certain substances while a mortar is a cup shaped object in which substances are ground therefore, the ground stones used were metates and their supplemental stones were manos. In order to create the depth, a ruler and a tape measure were used simultaneously; the ruler to show an even decline and the tape measure to show the distance. This process was fairly quick.

For the first metate, there was an obvious location where continual grinding had taken place. In order to grind on the surface, the side with a short width was rested upon a leg in order to make the top flat. The wheat berries were hard to grind and often rolled off the stone so it was easier and faster to stab them with the mano. It was concluded that grinding the wheat berries was easiest when their was a large pile of them on the metate instead of just a few.

For the second metate, there was also an obvious location where continual grinding had taken place. Since the metate was fairly consistent in height, it laid on the ground while used to grind. The quick cook barley was a lot easier to grind than the wheat berries and stabbing them as we did with the previous substance was actually harder to do so the barley was rubbed into the rock instead. We did not conclude which ground stone was easier to use.

SUMMARY

The ceramic station was rather limited in what could be stated and hypothesized since the potsherds did not make up a full ceramic item and the number of shards found was limited. In contrast, the lithic materials station held a vast variety of objects that even incomplete could be distinguished and missing parts could be deduced since the majority of the shape remained intact. Since the lithic materials station had a higher quantity and quality by the reasons above, the conclusions drawn would have more of an impact on larger areas and future items compared with the ceramic station. The ground stone station was more towards understanding methodology and use compared to analyzing physical structure and drawing conclusions about relationships between artifacts. As such, retrieving the viewed items from an archaeological dig would give the greatest indication about their stone and tool habits while providing insight into their pottery and domestic situation.

Lithic Materials 1, 2, 3, 4

Ceramic Materials 1, 2, 3

Ground Stone 1, 2

SOURCES

1.  Kelly, Robert and Thomas, David. Archaeology. Cengage Learning, 2016.

Essay: Stratigraphy Problem: Temple del Diablo

INTRODUCTION
A site map for Templo del Diablo was analyzed, looking at six house clusters labeled 1, 2, 3, 4, 5, and 6 with six trash pits A, B, C, D, E, and F and the two canals Eerie and Love. Two separate test trenches a to a’ and b to b’ were analyzed to gather data on relativity between the objects in the area. It is known that every house cluster contains six houses, was used for only one occupation phase, and used an unique trash pit. House clusters can be contemporaneous and the top of each feature in the test trenches corresponds to time that that feature was used. Lastly, it is known that the Love Canal is postdated by one occupation. These axioms were used in order to assess the information given to the full.

PROCEDURE
Gathering information from the test trenches, Table 1 was formulated. Further investigations of the map built Table 2, a reference of all features compared with their spatially associative features, relying upon the law of superposition, “the geological principle that in any pile of sedimentary rocks that have not been disturbed by folding or overturning, each bed is older than the layers above and younger than the layers below” (Kelly and Thomas, page 361). These tables were then compared to find the resulting location for each feature, drawn out in Table 3. Initially, Camron speculated on age while I recorded in a table format the relative dates and Liz wrote out the features that were not up to speculation. The level that each feature then belonged was placed using a combination of the information gathered. Afterwards, these speculations were discussed as a class.

OCCUPATIONS
Each housing cluster had its own occupation that was present for only one level. From the test trenches, it is known that level III had two trash pits meaning that two house clusters were contemporaneous at level III. Level IV contains house cluster 4 and trash pit C so these two are paired. It is given that the Love Canal is postdated by only one occupation. Since house cluster 5 is in level I and house cluster 6 is in level II, the Love Canal must be in level II in order to keep this statement true. Trash pit A is above the Love Canal which means that trash pit A is in level I, pairing it with house cluster 5. House cluster 3 is below the Love Canal in level II and above house cluster 4 in level IV so it must be in level III. Since trash pit D is above house clusters 3 and 2, house cluster 2 must be level III or below but house cluster 2 is above house cluster 1 which must be level IV or below since house cluster 1 is below trash pit E, house cluster 2 is also in level III. Trash pit D is above house clusters 2 and 3 in level III and below the Temple at level I so it must be in level II.

From here, it has been discovered that level I contains house cluster 5 and trash pit A, level II contains house cluster 6 and trash pit D, level III contains house clusters 2,3 and trash pits B,E, level IV contains house cluster 4 and trash pit C. It is known that house cluster 1 is level IV or below. Since there is no indication that level IV has contemporaneous house clusters, house cluster 1 must be in level V or VI with trash pit F. The map shows no indication of an unused level between house clusters 4 and 1. In addition, the flow of movement in each level also does not indicate a time period between occupations whereupon the area was not used. Thus, house cluster 1 with trash pit F are placed in level V, leaving level VI empty of occupation. Furthermore, in level three, house cluster 3 is paired with trash pit B and house cluster 2 is paired with trash pit E. Given that these house clusters are contemporaneous and the proximity between trash pit E and house cluster 2, it would not make sense for house cluster 3 to use a trash pit so close to house cluster 2 nor for house cluster 2 to use trash pit B when trash pit E is that much nearer. For house cluster 3, both trash pits are a distance away but trash pit B is closer than trash pit E so it also makes more sense to pair those two.

TEMPLE
On the test trench b to b’, the temple sits above level I and does not make an appearance in any of the levels below. However, on the stratigraphy map, The Love Canal and trash pit D are both shown as lower levels than the Temple. Since the Love Canal and trash pit D have both been placed in level II, the Temple must be level I or sooner. Since trash pit A and the Temple are both placed directly above the Love Canal, it is likely that the same occupation made both. If evidence of the Temple is found within trash pit A or house cluster 5, then their association would be known beyond doubt.

IRRIGATION CANALS

The Love Canal was placed in level II due to its relativity of other features on the map. It is obvious that trash pit A and the Temple both postdate the Love Canal and that the Love Canal is more recent than house cluster 3 or trash pit C. Given that the Love Canal is post dated by only one occupation, the Love Canal was largely beneficial in sorting the other components by level. In contrast, the Eerie Canal postdates house cluster 6 and trash pit B. Since test trench line a to a’ contains the Eerie Canal, the Eerie Canal is known to be in level I, trash pit B in level III and house cluster 6 in level II.

CONCLUSIONS

The map presented gives a lot of hidden information that can be deduced with careful observation and using the axioms given in the original set up. However, there are still a few degrees of uncertainty remaining that must be hypothesized from an archaeologist’s perspective in order to further deduce the relativity of all of the items presented. This assignment showcased the relations between four different groups and their stratigraphic relativities. Even with just four features; house clusters, trash pits, canals, the temple, how they all intersect and utilize the same spaces can be quite complicated.

Give an unlimited budget, only two more trenches would need to be completed to remove speculation about the levels of each feature. Let test trench c to c’ extend from house cluster 3 to house cluster 2, capturing trash pit D and the Love Canal. Then allow test trench d to d’ to extend from the Love Canal to trash pit F. Once these two trench lines are added, the level placement for all trash pits excluding A and all house clusters excluding 1 would be known as well as both canals. Since it is obvious that trash pit A and house cluster 1 do not conjoin, this extra information will add an extra degree of certainty to house clusters and trash pit locations that were deduced, particularly for house cluster 1 with trash pit F. Test trench c to c’ will also provide the level of the Love Canal with certainty which will make the observations made more concrete. Since house clusters 2,3 and trash pits B,E were all in level III, a more thorough analysis of their occupations could be conducted to prove which trash pit fit with which house cluster with more certainty than the analysis used. The only thing left to conclude would be the date of the Temple, which was already included in test trench b to b’ with limited assistance. A dating method on the materials used in the Temple’s formation could provide more feedback on its age with more certainty.

SOURCES

1.  Kelly, Robert and Thomas, David. Archaeology. Cengage Learning, 2016.

Essay: Sampling a Topographic Map

INTRODUCTION

On Friday, February 9th, an area was sampled randomly and judgmentally in order to find the twenty sites in the sample universe. The North and South had mountain topography while the middle of the area had a river in the shape of a Y stretching from the West to the East. The river was surrounded by a floodplain and the base of the mountain regions had foothills as well as a section to the West between the two branches of the river once outside the floodplains.

PROCEDURE

Using a number generator, 20 squares from the site universe were chosen and marked. Kacie wrote down the list of squares selected and inquired about sites within while I marked the squares chosen on the map and the sites found. Our group noted that the random sample found four small sites and one medium site, mainly in the area surrounding the river. Since the sites were not very varied in size and were spread across the sample universe in location, Camron suggested selecting one in every five squares, not repeating random sample squares in order to give each topographic type a chance at selection. Erin selected 81 as the starting point because our group hypothesized that the squares near the river bank would more likely contain a site. Kacie wrote down the numbers again and inquired about sites while I marked the squares chosen on the map and the sites found. Our group noted that the judgmental sample found three medium sites and one small site. This result added more depth to the previous understanding of the site universe but no large sites were located which limits the usefulness of the samples. After these two samples were charted, our group discussed the results. 

DATA AND RESULTS

At the end of both sample selections, 40 out of 100 squares had been tested and 45% of sites located. While none of the sites found were large, having near half of the sites recorded is enough to start testing hypotheses and drawing conclusions about the sample universe. With five small sites and four medium sites found out of a total of twenty sites of which nine are small, six medium, and five large, developed the sampling understanding of the sample universe. Our sampling efforts lead to a positive understanding of medium sites and their distribution and a fair understanding of the small sites given the amount of each found. However, since no large sites were discovered, how the large sites relate to the other sites as well as their locations and distributions remained unknown information.

The random sample found five of the sites; four small and one medium in size. Since the random sample used a number generator to get rid of bias, variability and pattern was left to the unknown as well. While the random sampling found five of the twenty sites, not many of the sample locations ended up being close to actual site locations and because each location has a 1% probability of being selected, there is little to guarantee finding sites.

The sample universe has a varied terrain and the sites found reflect that. While many sites from the random sample were found by the water source, these sites remained on the smaller size and could not properly reflect a dispersion of the sites, especially those of larger sizes. Based on what was gathered from the random sample, it would appear that the area is compromised of small sites near water and at the mountain base. The sample universe contains sites at varied locations and using a strongly methodical process without factoring in other elements of the sample universe such as topography can lead to a poor sampling of the region. The random sample views each of the squares equally regardless of location or probability within each topographic zone: the fact that the floodplains would have a higher chance of containing a site compared with the mountain region is simply not considered. That being said, the sites were varied across the sample universe and giving each square an equal chance at selection allowed for some of this variation to be expressed. Our random sample located sites in the floodplains and foothills in different regions of the map which was an indicator for location variation.

The judgmental sample found one fewer site compared to the random sample. With the random sample, out of 100 squares, each square had a 1% chance of being selected and a 20% chance of containing a site. There was a 5% chance that each selected square would contain a site. With the judgmental sample, the probability of a square becoming selected relied upon the other squares and their relation. Our group used a one in every five square method, modified slightly in order to not repeat squares from the random sample. In this way, the probability of a square containing a site was dependent upon the other squares selected. Using this method, the first selected square shifted the probabilities of the squares that were selected and the ones that were not. This method also made an assumption about the spacing and diversity of site settings which impacted the ability to locate squares with sites. Our judgmental sample was not more effective than the random sample but a different method could have been. A lot of the sites seem to be along borders of the different topographic zones: flood plains to foothills, foothills to mountains, river banks, and so forth. Concentrating a sample on squares that these border take place at would definitely have been more effective in locating sites. With twenty squares containing a site in the sample universe, variability is almost guaranteed and implementing a method that accounts for such is important. That being said, the sample universe shows certain trends with its samples, making it important to hypothesize potential patterns.

Selecting 20 out of 100 squares does not cover enough ground to give an accurate representation of the site selection. I would say that 33% to 50% would give a better understanding of the sample universe without being inefficient or sampling too many squares. Sampling all of the squares would be a huge waste of time and resources especially given that only 20% of the sample universe contains a site.

CONCLUSION

There are a multitude of methods for sampling a site with their own risks and benefits. Random sampling is a great starting sampling method since it does not make any assumptions on the sample universe before data is collected. However, random sampling treats every square with equal risk which can impact finding sites while sampling. Using judgmental sampling requires tact for the reasoning for selecting squares is much more individualized, creating higher risk. When it comes to archaeology, it appears the best method is oftentimes trial and error.